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# 11-4. Series and Their Notations

Series and Their Notations
• Use summation notation.
• Use the formula for the sum of the ﬁrst n terms of an arithmetic series.
• Use the formula for the sum of the ﬁrst n terms of a geometric series.
• Use the formula for the sum of an inﬁnite geometric series.
• Solve annuity problems.

# Using the Formula for Geometric Series

Just as the sum of the terms of an arithmetic sequence is called an arithmetic series, the sum of the terms in a geometric sequence is called a geometric series. Recall that a geometric sequence is a sequence in which the ratio of any two consecutive terms is the common ratio, <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>r</mi><mo>.</mo><mtext> </mtext></mrow></annotation-xml></semantics>[/itex]We can write the sum of the first <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>n</mi></mrow></annotation-xml></semantics>[/itex] terms of a geometric series as

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>S</mi></msub></mrow></annotation-xml></semantics>[/itex] n = a 1 +r a 1 + r 2 a 1 +...+ r n–1 a 1 .

Just as with arithmetic series, we can do some algebraic manipulation to derive a formula for the sum of the first<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>n</mi><mtext> </mtext></mrow></annotation-xml></semantics>[/itex]terms of a geometric series. We will begin by multiplying both sides of the equation by<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>r</mi><mo>.</mo><mtext> </mtext></mrow></annotation-xml></semantics>[/itex]

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>r</mi><msub/></mrow></annotation-xml></semantics>[/itex] S n =r a 1 + r 2 a 1 + r 3 a 1 +...+ r n a 1

Next, we subtract this equation from the original equation.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtable columnalign="left"><mtr><mtd><mrow/></mtd></mtr><mtr><mtd><mfrac><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mtext>    </mtext><msub/></mrow></mtd></mtr></mtable></mrow></mfrac></mtd></mtr></mtable></annotation-xml></semantics>[/itex] S n = a 1 +r a 1 + r 2 a 1 +...+ r n–1 a 1 −r S n =−(r a 1 + r 2 a 1 + r 3 a 1 +...+ r n a 1 ) (1−r) S n = a 1 − r n a 1

Notice that when we subtract, all but the first term of the top equation and the last term of the bottom equation cancel out. To obtain a formula for <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>S</mi></msub></mrow></annotation-xml></semantics>[/itex] n , divide both sides by <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>r</mi><mo stretchy="false">)</mo><mo>.</mo></mrow></annotation-xml></semantics>[/itex]

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>S</mi></msub></mrow></annotation-xml></semantics>[/itex] n = a 1 (1− r n ) 1−r  r≠1
Formula for the Sum of the First n Terms of a Geometric Series

A geometric series is the sum of the terms in a geometric sequence. The formula for the sum of the first<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>n</mi><mtext> </mtext></mrow></annotation-xml></semantics>[/itex]terms of a geometric sequence is represented as

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>S</mi></msub></mrow></annotation-xml></semantics>[/itex] n = a 1 (1− r n ) 1−r  r≠1
Given a geometric series, find the sum of the first n terms.
1. Identify<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msub/></mrow></annotation-xml></semantics>[/itex] a 1 , r, and n.
2. Substitute values for<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msub/></mrow></annotation-xml></semantics>[/itex] a 1 , r, and <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mi>n</mi></annotation-xml></semantics>[/itex] into the formula <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>S</mi></msub></mrow></annotation-xml></semantics>[/itex] n = a 1 (1– r n ) 1–r .
3. Simplify to find <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>S</mi></msub></mrow></annotation-xml></semantics>[/itex] n .
Finding the First n Terms of a Geometric Series

Use the formula to find the indicated partial sum of each geometric series.

1. <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>S</mi></msub></mrow></annotation-xml></semantics>[/itex] 11for the series<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> 8 + -4 + 2 + </mtext><mo>…</mo></mrow></annotation-xml></semantics>[/itex]
2. <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munderover><mrow><msup><mstyle displaystyle="true" mathsize="75%"><mo>∑</mo></mstyle></msup></mrow></munderover></mrow></annotation-xml></semantics>[/itex] ​ 6 k=1 3⋅ 2 k
1. <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>a</mi></msub></mrow></annotation-xml></semantics>[/itex] 1 =8, and we are given that <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>n</mi><mo>=</mo><mn>11.</mn></mrow></annotation-xml></semantics>[/itex]

We can find <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mi>r</mi></annotation-xml></semantics>[/itex] by dividing the second term of the series by the first.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>r</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics>[/itex] −4 8 =− 1 2

Substitute values for <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>a</mi></msub></mrow></annotation-xml></semantics>[/itex] 1 , r, and n into the formula and simplify.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><msub><mi>S</mi></msub></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics>[/itex] n = a 1 ( 1− r n ) 1−r S 11 = 8( 1− ( − 1 2 ) 11 ) 1−( − 1 2 ) ≈5.336
2. Find <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>a</mi></msub></mrow></annotation-xml></semantics>[/itex] 1 by substituting <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow></annotation-xml></semantics>[/itex] into the given explicit formula.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>a</mi></msub></mrow></annotation-xml></semantics>[/itex] 1 =3⋅ 2 1 =6

We can see from the given explicit formula that <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>r</mi><mo>=</mo><mn>2.</mn></mrow></annotation-xml></semantics>[/itex] The upper limit of summation is 6, so <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>n</mi><mo>=</mo><mn>6.</mn></mrow></annotation-xml></semantics>[/itex]

Substitute values for <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>a</mi></msub></mrow></annotation-xml></semantics>[/itex] 1 , r, and <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mi>n</mi></annotation-xml></semantics>[/itex] into the formula, and simplify.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><msub><mi>S</mi></msub></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics>[/itex] n = a 1 (1− r n ) 1−r S 6 = 6(1− 2 6 ) 1−2 =378

Use the formula to find the indicated partial sum of each geometric series.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>S</mi></msub></mrow></annotation-xml></semantics>[/itex] 20 for the series<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> 1,000 + 500 + 250 + </mtext><mo>…</mo></mrow></annotation-xml></semantics>[/itex]

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>≈</mo><mn>2</mn><mo>,</mo><mn>000.00</mn></mrow></annotation-xml></semantics>[/itex]

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mstyle displaystyle="true"><munderover><mo>∑</mo></munderover></mstyle></mrow></annotation-xml></semantics>[/itex] k=1 8 3 k

9,840

Solving an Application Problem with a Geometric Series

At a new job, an employee’s starting salary is 26,750. He receives a 1.6% annual raise. Find his total earnings at the end of 5 years. The problem can be represented by a geometric series with <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>a</mi></msub></mrow></annotation-xml></semantics>[/itex] 1 =26,750; <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>n</mi><mo>=</mo><mn>5</mn><mtext>;</mtext><mtext> </mtext></mrow></annotation-xml></semantics>[/itex]and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>r</mi><mo>=</mo><mn>1.016.</mn></mrow></annotation-xml></semantics>[/itex] Substitute values for<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msub/></mrow></annotation-xml></semantics>[/itex] a 1 , <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>r</mi><mtext>,</mtext></mrow></annotation-xml></semantics>[/itex] and <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>n</mi></mrow></annotation-xml></semantics>[/itex]into the formula and simplify to find the total amount earned at the end of 5 years. <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><msub><mi>S</mi></msub></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics>[/itex] n = a 1 (1− r n ) 1−r S 5 = 26,750(1− 1.016 5 ) 1−1.016 ≈138,099.03 He will have earned a total of138,099.03 by the end of 5 years.

At a new job, an employee’s starting salary is $32,100. She receives a 2% annual raise. How much will she have earned by the end of 8 years?$275,513.31

# Using the Formula for the Sum of an Infinite Geometric Series

Thus far, we have looked only at finite series. Sometimes, however, we are interested in the sum of the terms of an infinite sequence rather than the sum of only the first <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mi>n</mi></annotation-xml></semantics>[/itex]terms. An infinite series is the sum of the terms of an infinite sequence. An example of an infinite series is <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>2</mn><mo>+</mo><mn>4</mn><mo>+</mo><mn>6</mn><mo>+</mo><mn>8</mn><mo>+</mo><mn>...</mn></mrow></annotation-xml></semantics>[/itex]

This series can also be written in summation notation as <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mstyle displaystyle="true"><munderover><mo>∑</mo></munderover></mstyle></mrow></annotation-xml></semantics>[/itex] k=1 ∞ 2k, where the upper limit of summation is infinity. Because the terms are not tending to zero, the sum of the series increases without bound as we add more terms. Therefore, the sum of this infinite series is not defined. When the sum is not a real number, we say the series diverges.

## Determining Whether the Sum of an Infinite Geometric Series is Defined

If the terms of an infinite geometric series approach 0, the sum of an infinite geometric series can be defined. The terms in this series approach 0:

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>1</mn><mo>+</mo><mn>0.2</mn><mo>+</mo><mn>0.04</mn><mo>+</mo><mn>0.008</mn><mo>+</mo><mn>0.0016</mn><mo>+</mo><mn>...</mn></mrow></annotation-xml></semantics>[/itex]

The common ratio <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>r</mi><mtext> = 0</mtext><mtext>.2</mtext><mo>.</mo><mtext> </mtext></mrow></annotation-xml></semantics>[/itex] As<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>n</mi></mrow></annotation-xml></semantics>[/itex] gets very large, the values of <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mi>r</mi></msup></mrow></annotation-xml></semantics>[/itex] n get very small and approach 0. Each successive term affects the sum less than the preceding term. As each succeeding term gets closer to 0, the sum of the terms approaches a finite value. The terms of any infinite geometric series with <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>−</mo><mn>1</mn><mo><</mo><mi>r</mi><mo><</mo><mn>1</mn></mrow></annotation-xml></semantics>[/itex] approach 0; the sum of a geometric series is defined when <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>−</mo><mn>1</mn><mo><</mo><mi>r</mi><mo><</mo><mn>1.</mn></mrow></annotation-xml></semantics>[/itex]

Determining Whether the Sum of an Infinite Geometric Series is Defined

The sum of an infinite series is defined if the series is geometric and <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>−</mo><mn>1</mn><mo><</mo><mi>r</mi><mo><</mo><mn>1.</mn></mrow></annotation-xml></semantics>[/itex]

Given the first several terms of an infinite series, determine if the sum of the series exists.

1. Find the ratio of the second term to the first term.
2. Find the ratio of the third term to the second term.
3. Continue this process to ensure the ratio of a term to the preceding term is constant throughout. If so, the series is geometric.
4. If a common ratio, <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>r</mi><mo>,</mo></mrow></annotation-xml></semantics>[/itex] was found in step 3, check to see if <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>−</mo><mn>1</mn><mo><</mo><mi>r</mi><mo><</mo><mn>1</mn></mrow></annotation-xml></semantics>[/itex]. If so, the sum is defined. If not, the sum is not defined.
Determining Whether the Sum of an Infinite Series is Defined

Determine whether the sum of each infinite series is defined.

1. <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext>12 + 8 + 4 + </mtext><mo>…</mo></mrow></annotation-xml></semantics>[/itex]
2. <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mn>3</mn></mfrac></mrow></annotation-xml></semantics>[/itex] 4 + 1 2 + 1 3 +...
3. <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munderover><mstyle displaystyle="true"><mo>∑</mo></mstyle></munderover></mrow></annotation-xml></semantics>[/itex] k=1 ∞ 27⋅ ( 1 3 ) k
4. <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mstyle displaystyle="true"><munderover><mo>∑</mo></munderover></mstyle></mrow></annotation-xml></semantics>[/itex] k=1 ∞ 5k
1. The ratio of the second term to the first is <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mtext>2</mtext></mfrac></mrow></annotation-xml></semantics>[/itex] 3 , which is not the same as the ratio of the third term to the second, <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mn>1</mn></mfrac></mrow></annotation-xml></semantics>[/itex] 2 .The series is not geometric.
2. The ratio of the second term to the first is the same as the ratio of the third term to the second. The series is geometric with a common ratio of <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mn>2</mn></mfrac></mrow></annotation-xml></semantics>[/itex] 3 . The sum of the infinite series is defined.

3. The given formula is exponential with a base of <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mn>1</mn></mfrac></mrow></annotation-xml></semantics>[/itex] 3 ; the series is geometric with a common ratio of <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mn>1</mn></mfrac></mrow></annotation-xml></semantics>[/itex] 3 . The sum of the infinite series is defined.
4. The given formula is not exponential; the series is not geometric because the terms are increasing, and so cannot yield a finite sum.

Determine whether the sum of the infinite series is defined.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mn>1</mn></mfrac></mrow></annotation-xml></semantics>[/itex] 3 + 1 2 + 3 4 + 9 8 +...

The sum is defined. It is geometric.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>24</mn><mo>+</mo><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics>[/itex] −12 )+6+( −3 )+...

The sum of the infinite series is defined.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munderover><mstyle displaystyle="true" mathsize="140%"><mo>∑</mo></mstyle></munderover></mrow></annotation-xml></semantics>[/itex] k=1 ∞ 15⋅ (–0.3) k

The sum of the infinite series is defined.

## Finding Sums of Infinite Series

When the sum of an infinite geometric series exists, we can calculate the sum. The formula for the sum of an infinite series is related to the formula for the sum of the first <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mi>n</mi></annotation-xml></semantics>[/itex]terms of a geometric series.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>S</mi></msub></mrow></annotation-xml></semantics>[/itex] n = a 1 (1− r n ) 1−r

We will examine an infinite series with <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>r</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics>[/itex] 1 2 . What happens to <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mi>r</mi></msup></mrow></annotation-xml></semantics>[/itex] n as <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>n</mi></mrow></annotation-xml></semantics>[/itex] increases?

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtable columnalign="left"><mtr><mtd><msup><mrow><mo>(</mo></mrow></msup></mtd></mtr></mtable></annotation-xml></semantics>[/itex] 1 2 ) 2 = 1 4 ( 1 2 ) 3 = 1 8 ( 1 2 ) 4 = 1 16

The value of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msup/></mrow></annotation-xml></semantics>[/itex] r n  decreases rapidly. What happens for greater values of <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>n</mi><mo>?</mo></mrow></annotation-xml></semantics>[/itex]

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><msup><mrow><mo stretchy="false">(</mo><mfrac/></mrow></msup></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics>[/itex] 1 2 ) 10 = 1 1,024 ( 1 2 ) 20 = 1 1,048,576 ( 1 2 ) 30 = 1 1,073,741,824

As <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>n</mi></mrow></annotation-xml></semantics>[/itex] gets very large, <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mi>r</mi></msup></mrow></annotation-xml></semantics>[/itex] n gets very small. We say that, as <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>n</mi></mrow></annotation-xml></semantics>[/itex] increases without bound, <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mi>r</mi></msup></mrow></annotation-xml></semantics>[/itex] napproaches 0. As <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mi>r</mi></msup></mrow></annotation-xml></semantics>[/itex] n approaches 0,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>1</mn><mo>−</mo><msup/></mrow></annotation-xml></semantics>[/itex] r n approaches 1. When this happens, the numerator approaches<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msub/></mrow></annotation-xml></semantics>[/itex] a 1 . This give us a formula for the sum of an infinite geometric series.

Formula for the Sum of an Infinite Geometric Series

The formula for the sum of an infinite geometric series with <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>−1</mn><mo><</mo><mi>r</mi><mo><</mo><mn>1</mn></mrow></annotation-xml></semantics>[/itex] is

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>S</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics>[/itex] a 1 1−r

Given an infinite geometric series, find its sum.

1. Identify<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>a</mi></msub></mrow></annotation-xml></semantics>[/itex] 1and <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>r</mi><mo>.</mo></mrow></annotation-xml></semantics>[/itex]
2. Confirm that <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>–</mo><mn>1</mn><mo><</mo><mi>r</mi><mo><</mo><mn>1.</mn></mrow></annotation-xml></semantics>[/itex]
3. Substitute values for <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>a</mi></msub></mrow></annotation-xml></semantics>[/itex] 1 and <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mi>r</mi></annotation-xml></semantics>[/itex] into the formula, <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>S</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics>[/itex] a 1 1−r .
4. Simplify to find<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>S</mi><mo>.</mo></mrow></annotation-xml></semantics>[/itex]
Finding the Sum of an Infinite Geometric Series

Find the sum, if it exists, for the following:

1. <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>10</mn><mo>+</mo><mn>9</mn><mo>+</mo><mn>8</mn><mo>+</mo><mn>7</mn><mo>+</mo><mo>…</mo></mrow></annotation-xml></semantics>[/itex]
2. <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>248.6</mn><mo>+</mo><mn>99.44</mn><mo>+</mo><mn>39.776</mn><mo>+</mo><mtext> </mtext><mo>…</mo></mrow></annotation-xml></semantics>[/itex]
3. <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munderover><mstyle displaystyle="true"><mo>∑</mo></mstyle></munderover></mrow></annotation-xml></semantics>[/itex] k=1 ∞ 4,374⋅ (– 1 3 ) k–1
4. <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munderover><mstyle displaystyle="true"><mo>∑</mo></mstyle></munderover></mrow></annotation-xml></semantics>[/itex] k=1 ∞ 1 9 ⋅ ( 4 3 ) k
1. There is not a constant ratio; the series is not geometric.
2. There is a constant ratio; the series is geometric. <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>a</mi></msub></mrow></annotation-xml></semantics>[/itex] 1 =248.6and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>r</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics>[/itex] 99.44 248.6 =0.4, so the sum exists. Substitute <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>a</mi></msub></mrow></annotation-xml></semantics>[/itex] 1 =248.6 and <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>r</mi><mo>=</mo><mn>0.4</mn></mrow></annotation-xml></semantics>[/itex] into the formula and simplify to find the sum:

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>S</mi><mo>=</mo><mfrac/></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics>[/itex] a 1 1−r S= 248.6 1−0.4 =414. 3 ¯
3. The formula is exponential, so the series is geometric with <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>r</mi><mo>=</mo><mo>–</mo><mfrac/></mrow></annotation-xml></semantics>[/itex] 1 3 . Find<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>a</mi></msub></mrow></annotation-xml></semantics>[/itex] 1 by substituting <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow></annotation-xml></semantics>[/itex] into the given explicit formula:

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>a</mi></msub></mrow></annotation-xml></semantics>[/itex] 1 =4,374⋅ (– 1 3 ) 1–1 =4,374

Substitute <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>a</mi></msub></mrow></annotation-xml></semantics>[/itex] 1 =4,374 and <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>r</mi><mo>=</mo><mo>−</mo><mfrac/></mrow></annotation-xml></semantics>[/itex] 1 3 into the formula, and simplify to find the sum:

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>S</mi><mo>=</mo><mfrac/></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics>[/itex] a 1 1−r S= 4,374 1−(− 1 3 ) =3,280.5
4. The formula is exponential, so the series is geometric, but<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>r</mi><mo>></mo><mn>1.</mn><mtext> </mtext></mrow></annotation-xml></semantics>[/itex]The sum does not exist.
Finding an Equivalent Fraction for a Repeating Decimal

Find an equivalent fraction for the repeating decimal <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mn>0.</mn><mover accent="true"><mn>3</mn><mo>¯</mo></mover></annotation-xml></semantics>[/itex]

We notice the repeating decimal <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mn>0.</mn><mover accent="true"><mn>3</mn><mo>¯</mo></mover><mo>=</mo><mn>0.333...</mn></annotation-xml></semantics>[/itex] so we can rewrite the repeating decimal as a sum of terms.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mn>0.</mn><mover accent="true"><mn>3</mn><mo>¯</mo></mover><mo>=</mo><mn>0.3</mn><mo>+</mo><mn>0.03</mn><mo>+</mo><mn>0.003</mn><mo>+</mo><mn>...</mn></annotation-xml></semantics>[/itex]

Looking for a pattern, we rewrite the sum, noticing that we see the first term multiplied to 0.1 in the second term, and the second term multiplied to 0.1 in the third term.

Notice the pattern; we multiply each consecutive term by a common ratio of 0.1 starting with the first term of 0.3. So, substituting into our formula for an infinite geometric sum, we have

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>S</mi></msub></mrow></annotation-xml></semantics>[/itex] n = a 1 1−r = 0.3 1−0.1 = 0.3 0.9 = 1 3 .

Find the sum, if it exists.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>2</mn><mo>+</mo><mfrac/></mrow></annotation-xml></semantics>[/itex] 2 3 + 2 9 +...

3

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mstyle displaystyle="true"><munderover><mo>∑</mo></munderover></mstyle></mrow></annotation-xml></semantics>[/itex] k=1 ∞ 0.76k+1

The series is not geometric.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mstyle displaystyle="true"><munderover><mo>∑</mo></munderover></mstyle></mrow></annotation-xml></semantics>[/itex] k=1 ∞ ( − 3 8 ) k

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>−</mo><mfrac/></mrow></annotation-xml></semantics>[/itex] 3 11

# Solving Annuity Problems

At the beginning of the section, we looked at a problem in which a couple invested a set amount of money each month into a college fund for six years. An annuity is an investment in which the purchaser makes a sequence of periodic, equal payments. To find the amount of an annuity, we need to find the sum of all the payments and the interest earned. In the example, the couple invests $50 each month. This is the value of the initial deposit. The account paid 6% annual interest, compounded monthly. To find the interest rate per payment period, we need to divide the 6% annual percentage interest (APR) rate by 12. So the monthly interest rate is 0.5%. We can multiply the amount in the account each month by 100.5% to find the value of the account after interest has been added. We can find the value of the annuity right after the last deposit by using a geometric series with <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>a</mi></msub></mrow></annotation-xml></semantics>[/itex] 1 =50 and <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>r</mi><mo>=</mo><mn>100.5</mn><mi>%</mi><mo>=</mo><mn>1.005.</mn></mrow></annotation-xml></semantics>[/itex] After the first deposit, the value of the annuity will be$50. Let us see if we can determine the amount in the college fund and the interest earned.

We can find the value of the annuity after <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mi>n</mi></annotation-xml></semantics>[/itex] deposits using the formula for the sum of the first <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mi>n</mi></annotation-xml></semantics>[/itex] terms of a geometric series. In 6 years, there are 72 months, so <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>n</mi><mo>=</mo><mn>72.</mn></mrow></annotation-xml></semantics>[/itex] We can substitute <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>a</mi></msub></mrow></annotation-xml></semantics>[/itex] 1 =50, r=1.005, and n=72 into the formula, and simplify to find the value of the annuity after 6 years.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>S</mi></msub></mrow></annotation-xml></semantics>[/itex] 72 = 50(1− 1.005 72 ) 1−1.005 ≈4,320.44

After the last deposit, the couple will have a total of $4,320.44 in the account. Notice, the couple made 72 payments of$50 each for a total of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mtext>72(50) = $3,600</mtext><mtext>.</mtext><mtext> </mtext></mrow></annotation-xml></semantics>[/itex]This means that because of the annuity, the couple earned$720.44 interest in their college fund.

Given an initial deposit and an interest rate, find the value of an annuity.

1. Determine<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msub/></mrow></annotation-xml></semantics>[/itex] a 1 , the value of the initial deposit.
2. Determine<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>n</mi><mtext>,</mtext><mtext> </mtext></mrow></annotation-xml></semantics>[/itex]the number of deposits.
3. Determine<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>r</mi><mo>.</mo></mrow></annotation-xml></semantics>[/itex]
1. Divide the annual interest rate by the number of times per year that interest is compounded.
2. Add 1 to this amount to find <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>r</mi><mo>.</mo></mrow></annotation-xml></semantics>[/itex]
4. Substitute values for<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msub/></mrow></annotation-xml></semantics>[/itex] a 1 , r, and n  into the formula for the sum of the first <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mi>n</mi></annotation-xml></semantics>[/itex] terms of a geometric series,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>S</mi></msub></mrow></annotation-xml></semantics>[/itex] n = a 1 (1– r n ) 1–r .
5. Simplify to find <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>S</mi></msub></mrow></annotation-xml></semantics>[/itex] n , the value of the annuity after <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mi>n</mi></annotation-xml></semantics>[/itex] deposits.
Solving an Annuity Problem

A deposit of $100 is placed into a college fund at the beginning of every month for 10 years. The fund earns 9% annual interest, compounded monthly, and paid at the end of the month. How much is in the account right after the last deposit? The value of the initial deposit is$100, so<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msub/></mrow></annotation-xml></semantics>[/itex] a 1 =100. A total of 120 monthly deposits are made in the 10 years, so<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>n</mi><mo>=</mo><mn>120.</mn></mrow></annotation-xml></semantics>[/itex] To find <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>r</mi><mo>,</mo><mtext> </mtext></mrow></annotation-xml></semantics>[/itex]divide the annual interest rate by 12 to find the monthly interest rate and add 1 to represent the new monthly deposit.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>r</mi><mo>=</mo><mn>1</mn><mo>+</mo><mfrac/></mrow></annotation-xml></semantics>[/itex] 0.09 12 =1.0075

Substitute<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msub/></mrow></annotation-xml></semantics>[/itex] a 1 =100, r=1.0075, and n=120 into the formula for the sum of the first <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mi>n</mi></annotation-xml></semantics>[/itex] terms of a geometric series, and simplify to find the value of the annuity.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>S</mi></msub></mrow></annotation-xml></semantics>[/itex] 120 = 100(1− 1.0075 120 ) 1−1.0075 ≈19,351.43

So the account has $19,351.43 after the last deposit is made. At the beginning of each month,$200 is deposited into a retirement fund. The fund earns 6% annual interest, compounded monthly, and paid into the account at the end of the month. How much is in the account if deposits are made for 10 years?

$92,408.18 Access these online resources for additional instruction and practice with series. # Key Equations  sum of the firstn[/itex] terms of an arithmetic series S[/itex] n = n( a 1 + a n ) 2 sum of the firstn[/itex] terms of a geometric series S[/itex] n = a 1 (1− r n ) 1−r ⋅r≠1 sum of an infinite geometric series with1<r< 1[/itex] S[/itex] n = a 1 1−r ⋅r≠1 # Key Concepts • The sum of the terms in a sequence is called a series. • A common notation for series is called summation notation, which uses the Greek letter sigma to represent the sum. See [link]. • The sum of the terms in an arithmetic sequence is called an arithmetic series. • The sum of the first<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mi>n</mi></annotation-xml></semantics>[/itex]terms of an arithmetic series can be found using a formula. See [link] and [link]. • The sum of the terms in a geometric sequence is called a geometric series. • The sum of the first<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mi>n</mi></annotation-xml></semantics>[/itex]terms of a geometric series can be found using a formula. See [link] and [link]. • The sum of an infinite series exists if the series is geometric with <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>–1</mn><mo><</mo><mi>r</mi><mo><</mo><mn>1.</mn></mrow></annotation-xml></semantics>[/itex] • If the sum of an infinite series exists, it can be found using a formula. See [link], [link], and [link]. • An annuity is an account into which the investor makes a series of regularly scheduled payments. The value of an annuity can be found using geometric series. See [link]. # Section Exercises ## Verbal What is an <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>n</mi><mtext>th</mtext></mrow></annotation-xml></semantics>[/itex] partial sum? An <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>n</mi><mtext>th</mtext></mrow></annotation-xml></semantics>[/itex] partial sum is the sum of the first <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mi>n</mi></annotation-xml></semantics>[/itex] terms of a sequence. What is the difference between an arithmetic sequence and an arithmetic series? What is a geometric series? A geometric series is the sum of the terms in a geometric sequence. How is finding the sum of an infinite geometric series different from finding the <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>n</mi><mtext>th</mtext></mrow></annotation-xml></semantics>[/itex] partial sum? What is an annuity? An annuity is a series of regular equal payments that earn a constant compounded interest. ## Algebraic For the following exercises, express each description of a sum using summation notation. The sum of terms <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mi>m</mi></msup></mrow></annotation-xml></semantics>[/itex] 2 +3m from <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>m</mi><mo>=</mo><mn>1</mn></mrow></annotation-xml></semantics>[/itex] to <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>m</mi><mo>=</mo><mn>5</mn></mrow></annotation-xml></semantics>[/itex] The sum from of <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>n</mi><mo>=</mo><mn>0</mn></mrow></annotation-xml></semantics>[/itex] to <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>n</mi><mo>=</mo><mn>4</mn></mrow></annotation-xml></semantics>[/itex] of <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>5</mn><mi>n</mi></mrow></annotation-xml></semantics>[/itex] <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mstyle displaystyle="true"><munderover><mo>∑</mo></munderover></mstyle></mrow></annotation-xml></semantics>[/itex] n=0 4 5n The sum of <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>6</mn><mi>k</mi><mo>−</mo><mn>5</mn></mrow></annotation-xml></semantics>[/itex] from <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>k</mi><mo>=</mo><mo>−</mo><mn>2</mn></mrow></annotation-xml></semantics>[/itex] to <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow></annotation-xml></semantics>[/itex] The sum that results from adding the number 4 five times <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mstyle displaystyle="true"><munderover><mo>∑</mo></munderover></mstyle></mrow></annotation-xml></semantics>[/itex] k=1 5 4 For the following exercises, express each arithmetic sum using summation notation. <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>5</mn><mo>+</mo><mn>10</mn><mo>+</mo><mn>15</mn><mo>+</mo><mn>20</mn><mo>+</mo><mn>25</mn><mo>+</mo><mn>30</mn><mo>+</mo><mn>35</mn><mo>+</mo><mn>40</mn><mo>+</mo><mn>45</mn><mo>+</mo><mn>50</mn></mrow></annotation-xml></semantics>[/itex] <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>10</mn><mo>+</mo><mn>18</mn><mo>+</mo><mn>26</mn><mo>+</mo><mo>…</mo><mo>+</mo><mn>162</mn></mrow></annotation-xml></semantics>[/itex] <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mstyle displaystyle="true"><munderover><mo>∑</mo></munderover></mstyle></mrow></annotation-xml></semantics>[/itex] k=1 20 8k+2 <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mn>1</mn></mfrac></mrow></annotation-xml></semantics>[/itex] 2 +1+ 3 2 +2+…+4 For the following exercises, use the formula for the sum of the first <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mi>n</mi></annotation-xml></semantics>[/itex] terms of each arithmetic sequence. <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mn>3</mn></mfrac></mrow></annotation-xml></semantics>[/itex] 2 +2+ 5 2 +3+ 7 2 <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>S</mi></msub></mrow></annotation-xml></semantics>[/itex] 5 = 5( 3 2 + 7 2 ) 2 <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>19</mn><mo>+</mo><mn>25</mn><mo>+</mo><mn>31</mn><mo>+</mo><mo>…</mo><mo>+</mo><mn>73</mn></mrow></annotation-xml></semantics>[/itex] <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>3.2</mn><mo>+</mo><mn>3.4</mn><mo>+</mo><mn>3.6</mn><mo>+</mo><mo>…</mo><mo>+</mo><mn>5.6</mn></mrow></annotation-xml></semantics>[/itex] <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>S</mi></msub></mrow></annotation-xml></semantics>[/itex] 13 = 13( 3.2+5.6 ) 2 For the following exercises, express each geometric sum using summation notation. <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>1</mn><mo>+</mo><mn>3</mn><mo>+</mo><mn>9</mn><mo>+</mo><mn>27</mn><mo>+</mo><mn>81</mn><mo>+</mo><mn>243</mn><mo>+</mo><mn>729</mn><mo>+</mo><mn>2187</mn></mrow></annotation-xml></semantics>[/itex] <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>8</mn><mo>+</mo><mn>4</mn><mo>+</mo><mn>2</mn><mo>+</mo><mo>…</mo><mo>+</mo><mn>0.125</mn></mrow></annotation-xml></semantics>[/itex] <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mstyle displaystyle="true"><munderover><mo>∑</mo></munderover></mstyle></mrow></annotation-xml></semantics>[/itex] k=1 7 8⋅ 0.5 k−1 <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>−</mo><mfrac/></mrow></annotation-xml></semantics>[/itex] 1 6 + 1 12 − 1 24 +…+ 1 768 For the following exercises, use the formula for the sum of the first <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>n</mi></mrow></annotation-xml></semantics>[/itex] terms of each geometric sequence, and then state the indicated sum. <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>9</mn><mo>+</mo><mn>3</mn><mo>+</mo><mn>1</mn><mo>+</mo><mfrac/></mrow></annotation-xml></semantics>[/itex] 1 3 + 1 9 <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>S</mi></msub></mrow></annotation-xml></semantics>[/itex] 5 = 9( 1− ( 1 3 ) 5 ) 1− 1 3 = 121 9 ≈13.44 <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mstyle displaystyle="true"><munderover><mo>∑</mo></munderover></mstyle></mrow></annotation-xml></semantics>[/itex] n=1 9 5⋅ 2 n−1 <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mstyle displaystyle="true"><munderover><mo>∑</mo></munderover></mstyle></mrow></annotation-xml></semantics>[/itex] a=1 11 64⋅ 0.2 a−1 <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>S</mi></msub></mrow></annotation-xml></semantics>[/itex] 11 = 64( 1− 0.2 11 ) 1−0.2 = 781,249,984 9,765,625 ≈80 For the following exercises, determine whether the infinite series has a sum. If so, write the formula for the sum. If not, state the reason. <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>12</mn><mo>+</mo><mn>18</mn><mo>+</mo><mn>24</mn><mo>+</mo><mn>30</mn><mo>+</mo><mn>...</mn></mrow></annotation-xml></semantics>[/itex] <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>2</mn><mo>+</mo><mn>1.6</mn><mo>+</mo><mn>1.28</mn><mo>+</mo><mn>1.024</mn><mo>+</mo><mn>...</mn></mrow></annotation-xml></semantics>[/itex] The series is defined. <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>S</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics>[/itex] 2 1−0.8 <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mstyle displaystyle="true"><munderover><mo>∑</mo></munderover></mstyle></mrow></annotation-xml></semantics>[/itex] m=1 ∞ 4 m−1 <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munderover><mrow><msup><mstyle displaystyle="true" mathsize="140%"><mo>∑</mo></mstyle></msup></mrow></munderover></mrow></annotation-xml></semantics>[/itex] ​ ∞ k=1 − ( − 1 2 ) k−1 The series is defined. <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>S</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics>[/itex] −1 1−( − 1 2 ) ## Graphical For the following exercises, use the following scenario. Javier makes monthly deposits into a savings account. He opened the account with an initial deposit of$50. Each month thereafter he increased the previous deposit amount by $20. Graph the arithmetic sequence showing one year of Javier’s deposits. Graph the arithmetic series showing the monthly sums of one year of Javier’s deposits. For the following exercises, use the geometric series<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mrow><mstyle displaystyle="true"><munderover><mo>∑</mo></munderover></mstyle></mrow></msup></mrow></annotation-xml></semantics>[/itex] k=1 ∞ ( 1 2 ) k . Graph the first 7 partial sums of the series. What number does <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>S</mi></msub></mrow></annotation-xml></semantics>[/itex] n seem to be approaching in the graph? Find the sum to explain why this makes sense. Sample answer: The graph of <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>S</mi></msub></mrow></annotation-xml></semantics>[/itex] n seems to be approaching 1. This makes sense because<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mstyle displaystyle="true"><munderover><mo>∑</mo></munderover></mstyle></mrow></annotation-xml></semantics>[/itex] k=1 ∞ ( 1 2 ) kis a defined infinite geometric series with <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>S</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics>[/itex] 1 2 1–( 1 2 ) =1. ## Numeric For the following exercises, find the indicated sum. <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mstyle displaystyle="true"><munderover><mo>∑</mo></munderover></mstyle></mrow></annotation-xml></semantics>[/itex] a=1 14 a <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mstyle displaystyle="true"><munderover><mo>∑</mo></munderover></mstyle></mrow></annotation-xml></semantics>[/itex] n=1 6 n(n−2) 49 <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mstyle displaystyle="true"><munderover><mo>∑</mo></munderover></mstyle></mrow></annotation-xml></semantics>[/itex] k=1 17 k 2 <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mstyle displaystyle="true"><munderover><mo>∑</mo></munderover></mstyle></mrow></annotation-xml></semantics>[/itex] k=1 7 2 k 254 For the following exercises, use the formula for the sum of the first <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mi>n</mi></annotation-xml></semantics>[/itex] terms of an arithmetic series to find the sum. <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>−</mo><mn>1.7</mn><mo>+</mo><mo>−</mo><mn>0.4</mn><mo>+</mo><mn>0.9</mn><mo>+</mo><mn>2.2</mn><mo>+</mo><mn>3.5</mn><mo>+</mo><mn>4.8</mn></mrow></annotation-xml></semantics>[/itex] <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>6</mn><mo>+</mo><mfrac/></mrow></annotation-xml></semantics>[/itex] 15 2 +9+ 21 2 +12+ 27 2 +15 <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>S</mi></msub></mrow></annotation-xml></semantics>[/itex] 7 = 147 2 <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>−</mo><mn>1</mn><mo>+</mo><mn>3</mn><mo>+</mo><mn>7</mn><mo>+</mo><mn>...</mn><mo>+</mo><mn>31</mn></mrow></annotation-xml></semantics>[/itex] <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mstyle displaystyle="true"><munderover><mo>∑</mo></munderover></mstyle></mrow></annotation-xml></semantics>[/itex] k=1 11 ( k 2 − 1 2 ) <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>S</mi></msub></mrow></annotation-xml></semantics>[/itex] 11 = 55 2 For the following exercises, use the formula for the sum of the first <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mi>n</mi></annotation-xml></semantics>[/itex] terms of a geometric series to find the partial sum. <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>S</mi></msub></mrow></annotation-xml></semantics>[/itex] 6 for the series <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>−</mo><mn>2</mn><mo>−</mo><mn>10</mn><mo>−</mo><mn>50</mn><mo>−</mo><mn>250...</mn></mrow></annotation-xml></semantics>[/itex] <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>S</mi></msub></mrow></annotation-xml></semantics>[/itex] 7 for the series <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>0.4</mn><mo>−</mo><mn>2</mn><mo>+</mo><mn>10</mn><mo>−</mo><mn>50...</mn></mrow></annotation-xml></semantics>[/itex] <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>S</mi></msub></mrow></annotation-xml></semantics>[/itex] 7 =5208.4 <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mstyle displaystyle="true"><munderover><mo>∑</mo></munderover></mstyle></mrow></annotation-xml></semantics>[/itex] k=1 9 2 k−1 <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mstyle displaystyle="true"><munderover><mo>∑</mo></munderover></mstyle></mrow></annotation-xml></semantics>[/itex] n=1 10 −2⋅ ( 1 2 ) n−1 <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>S</mi></msub></mrow></annotation-xml></semantics>[/itex] 10 =− 1023 256 For the following exercises, find the sum of the infinite geometric series. <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>4</mn><mo>+</mo><mn>2</mn><mo>+</mo><mn>1</mn><mo>+</mo><mfrac/></mrow></annotation-xml></semantics>[/itex] 1 2 ... <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>−</mo><mn>1</mn><mo>−</mo><mfrac/></mrow></annotation-xml></semantics>[/itex] 1 4 − 1 16 − 1 64 ... <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>S</mi><mo>=</mo><mo>−</mo><mfrac/></mrow></annotation-xml></semantics>[/itex] 4 3 <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munderover><mrow><msup><mstyle displaystyle="true" mathsize="140%"><mo>∑</mo></mstyle></msup></mrow></munderover></mrow></annotation-xml></semantics>[/itex] ​ ∞ k=1 3⋅ ( 1 4 ) k−1 <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mstyle displaystyle="true"><munderover><mo>∑</mo></munderover></mstyle></mrow></annotation-xml></semantics>[/itex] n=1 ∞ 4.6⋅ 0.5 n−1 <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>S</mi><mo>=</mo><mn>9.2</mn></mrow></annotation-xml></semantics>[/itex] For the following exercises, determine the value of the annuity for the indicated monthly deposit amount, the number of deposits, and the interest rate. Deposit amount: <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext>$</mtext><mn>50</mn><mo>;</mo></mrow></annotation-xml></semantics>[/itex] total deposits: <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>60</mn><mo>;</mo></mrow></annotation-xml></semantics>[/itex] interest rate: <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>5</mn><mi>%</mi><mo>,</mo></mrow></annotation-xml></semantics>[/itex] compounded monthly

Deposit amount: <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext>$</mtext><mn>150</mn><mo>;</mo></mrow></annotation-xml></semantics>[/itex] total deposits: <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>24</mn><mo>;</mo></mrow></annotation-xml></semantics>[/itex] interest rate: <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>3</mn><mi>%</mi><mo>,</mo></mrow></annotation-xml></semantics>[/itex] compounded monthly$3,705.42

Deposit amount: <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext>$</mtext><mn>450</mn><mo>;</mo></mrow></annotation-xml></semantics>[/itex] total deposits: <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>60</mn><mo>;</mo></mrow></annotation-xml></semantics>[/itex] interest rate: <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>4.5</mn><mi>%</mi><mo>,</mo></mrow></annotation-xml></semantics>[/itex] compounded quarterly Deposit amount: <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext>$</mtext><mn>100</mn><mo>;</mo></mrow></annotation-xml></semantics>[/itex] total deposits: <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>120</mn><mo>;</mo></mrow></annotation-xml></semantics>[/itex] interest rate: <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>10</mn><mi>%</mi><mo>,</mo></mrow></annotation-xml></semantics>[/itex] compounded semi-annually

## Glossary

annuity
an investment in which the purchaser makes a sequence of periodic, equal payments
arithmetic series
the sum of the terms in an arithmetic sequence
diverge
a series is said to diverge if the sum is not a real number
geometric series
the sum of the terms in a geometric sequence
index of summation
in summation notation, the variable used in the explicit formula for the terms of a series and written below the sigma with the lower limit of summation
infinite series
the sum of the terms in an infinite sequence
lower limit of summation
the number used in the explicit formula to find the first term in a series
nth partial sum
the sum of the first<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mi>n</mi></annotation-xml></semantics>[/itex]terms of a sequence
series
the sum of the terms in a sequence
summation notation
a notation for series using the Greek letter sigma; it includes an explicit formula and specifies the first and last terms in the series
upper limit of summation
the number used in the explicit formula to find the last term in a series