
8.2: Geometric Series

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Skills to Develop

In this section, we strive to understand the ideas generated by the following important questions:

• What is a geometric series?
• What is a partial sum of a geometric series?
• What is a simplified form of the nth partial sum of a geometric series?
• Under what conditions does a geometric series converge?
• What is the sum of a convergent geometric series?

Many important sequences are generated through the process of addition. In Preview Activity $$\PageIndex{1}$$, we see a particular example of a special type of sequence that is connected to a sum.

Preview Activity $$\PageIndex{1}$$

Warfarin is an anticoagulant that prevents blood clotting; often it is prescribed to stroke victims in order to help ensure blood flow. The level of warfarin has to reach a certain concentration in the blood in order to be effective. Suppose warfarin is taken by a particular patient in a 5 mg dose each day. The drug is absorbed by the body and some is excreted from the system between doses. Assume that at the end of a 24 hour period, 8% of the drug remains in the body. Let $$Q(n)$$ be the amount (in mg) of warfarin in the body before the $$(n + 1)$$st dose of the drug is administered.

1. Explain why $$Q(1) = 5 \times 0.08$$ mg.
2. Explain why $$Q(2) = (5 + Q(1)) \times 0.08$$ mg. Then show that $$Q(2) = (5 \times 0.08) (1 + 0.08)$$ mg.
3. Explain why $$Q(3) = (5 + Q(2)) \times 0.08$$ mg. Then show that $$Q(3) = (5 \times 0.08) 1 + 0.08 + 0.082$$ mg.
4. Explain why $$Q(4) = (5 + Q(3)) \times 0.08$$ mg. Then show that $$Q(4) = (5 \times 0.08) 1 + 0.08 + 0.082 + 0.083$$ mg.
5. There is a pattern that you should see emerging. Use this pattern to find a formula for $$Q(n)$$, where $$n$$ is an arbitrary positive integer.
6. Complete Table 8.2 with values of $$Q(n)$$ for the provided $$n$$-values (reporting $$Q(n)$$ to 10 decimal places). What appears to be happening to the sequence $$Q(n)$$ as $$n$$ increases?
 $$Q(1)$$ 0.40 $$Q(2)$$ $$Q(3)$$ $$Q(4)$$ $$Q(5)$$ $$Q(6)$$ $$Q(7)$$ $$Q(8)$$ $$Q(9)$$ $$Q(10)$$

Table 8.2: Values of $$Q(n)$$ for selected values of $$n$$

Geometric Sums

In Preview Activity 8.2 we encountered the sum

$(5 \times 0.08) 1 + 0.08 + 0.082 + 0.083 + · · · + 0.08^{n−1}. \nonumber$

In order to evaluate the long-term level of Warfarin in the patient’s system, we will want to fully understand the sum in this expression. This sum has the form

$a + ar + ar^2 + · · · + ar^{n−1} \tag{8.1}\label{8.1}$

where $$a = 5 \times 0.08$$ and $$r = 0.08$$. Such a sum is called a geometric sum with ratio r. We will analyze this sum in more detail in the next activity.

Activity $$\PageIndex{1}$$

Let $$a$$ and $$r$$ be real numbers (with $$r \neq 1$$) and let

$S_n = a + ar + ar^2 + · · · + ar^{n−1}. \nonumber$

In this activity we will find a shortcut formula for $$S_n$$ that does not involve a sum of $$n$$ terms.

1. Multiply $$S_n$$ by $$r$$. What does the resulting sum look like?
2. Subtract $$r S_n$$ from $$S_n$$ and explain why

$S_n − r S_n = a − ar^n .\tag{8.2}\label{8.2}$

c.  Solve Equation ($$\ref{8.2}$$) for $$S_n$$ to find a simple formula for $$S_n$$ that does not involve adding $$n$$ terms.

We can summarize the result of Activity $$\PageIndex{1}$$ in the following way. A geometric sum Sn is a sum of the form

$Sn = a + ar + ar^2 + · · · + ar^{n−1} , \tag{8.3}\label{8.3}$

where $$a$$ and $$r$$ are real numbers such that $$r \neq 1$$. The geometric sum $$S_n$$ can be written more simply as

$Sn = a + ar + ar^2 + · · · + ar^n−1 = a(1 − r^n ) 1 − r . \tag{8.4}\label{8.4}$

We now apply Equation $$\ref{8.4}$$ to the example involving warfarin from Preview Activity 8.2. Recall that

$$Q(n) = (5 \times 0.08) 1 + 0.08 + 0.082 + 0.083 + · · · + 0.08^{n−1}$$ mg,

so $$Q(n)$$ is a geometric sum with $$a = 5 \times 0.08 = 0.4$$ and $$r = 0.08$$. Thus,

$Q(n) = 0.4 1 − 0.08n 1 − 0.08 ! = 1 2.3 (1 − 0.08n ) . \nonumber$

Notice that as $$n$$ goes to infinity, the value of $$0.08^n$$ goes to $$0$$. So,

$\lim_{n \rightarrow \infty}Q(n) = \lim_{n \rightarrow \infty}\dfrac{1}{2.3}(1-0.08^n)= \dfrac{1}{2.3}\approx 0.435. \nonumber$

Therefore, the long-term level of Warfarin in the blood under these conditions is $$\dfrac{1}{2.3}$$, which is approximately $$0.435$$ mg. To determine the long-term effect of Warfarin, we considered a geometric sum of $$n$$ terms, and then considered what happened as $$n$$ was allowed to grow without bound. In this sense, we were actually interested in an infinite geometric sum (the result of letting n go to infinity in the finite sum). We call such an infinite geometric sum a geometric series.

Definition

A geometric series is an infinite sum of the form

$a +ar +ar^2 + ... = \sum_{n=0}^\infty ar^n. \tag{8.5}\label{8.5}$

The value of $$r$$ in the geometric series in Equation $$\ref{8.5}$$ is called the common ratio of the series because the ratio of the $$(n + 1)$$st term $$ar^n$$ to the $$n$$th term $$ar^{n−1}$$ is always $$r$$.

Geometric series are very common in mathematics and arise naturally in many different situations. As a familiar example, suppose we want to write the number with repeating decimal expansion

$N = 0.1212\overline{12} \nonumber$

as a rational number.

Observe that

$N = 0.12 + 0.0012 + 0.000012 + · · · \nonumber$

$= \dfrac{12}{100}+\dfrac{12}{100}\dfrac{1}{100}+ \dfrac{12}{100}\dfrac{1}{100}^2 + ... , \nonumber$

which is an infinite geometric series with $$a = \dfrac{12}{100}$$ and $$r = \dfrac{1}{100}$$. In the same way that we were able to find a shortcut formula for the value of a (finite) geometric sum, we would like to develop a formula for the value of a (infinite) geometric series. We explore this idea in the following activity.

Activity $$\PageIndex{2}$$

Let $$r \neq 1$$ and $$a$$ be real numbers and let

$S = a + ar + ar^2 + · · · ar^{n−1} + · · · \nonumber$

be an infinite geometric series. For each positive integer $$n$$, let

$S_n = a + ar + ar^2 + · · · + ar^{n−1}. \nonumber$

Recall that

$S_n = a \dfrac{1-r^n}{1-r}. \nonumber$

a. What should we allow n to approach in order to have $$S_n$$ approach $$S$$

b. What is the value of $$\lim_{n \rightarrow \infty} r^n$$ for

• |$$r$$| > 1?
• |$$r$$| < 1?

Explain.

c. If |$$r$$| < 1, use the formula for $$S_n$$ and your observations in (a) and (b) to explain why $$S$$ is finite and find a resulting formula for $$S$$.

From our work in Activity 8.5, we can now find the value of the geometric series $$N = \dfrac{12} {100} + \dfrac{12}{100} \dfrac{1}{100} + \dfrac{12}{100} \dfrac{1}{100}^2+ · · ·$$. In particular, using $$a = \dfrac{12}{100}$$ and $$r = \frac{1}{100}$$, we see that

$$N = \dfrac{12}{100} \left(\dfrac{1}{1-\dfrac{1}{100}}\right) = \dfrac{12}{100} \left(\dfrac{100} {99} \right) = \dfrac{4}{33}$$.

It is important to notice that a geometric sum is simply the sum of a finite number of terms of a geometric series. In other words, the geometric sum $$S_n$$ for the geometric series

$\sum_{k=0}^\infty ar^k \nonumber$

is

$S_n = a + ar + ar^2 +....ar^{n-1}= \sum_{k=0}^{n-1} ar^k \nonumber$

We also call this sum $$S_n$$ the $$n$$th partial sum of the geometric series. We summarize our recent work with geometric series as follows.

• A geometric series is an infinite sum of the form

$a + ar +ar^2 + ... = \sum_{n=0}^\infty ar^n, \tag{8.6}\label{8.6}$

where $$a$$ and $$r$$ are real numbers such that $$r \neq 0$$.

• The nth partial sum Sn of the geometric series is

$S_n = a + ar + ar^2 + · · · + ar^{n−1} . \nonumber$

• If |$$r$$| < 1, then using the fact that $$S_n = a \dfrac{1−r^n}{1−r}$$, it follows that the sum $$S$$ of the geometric series ($$\ref{8.6}$$) is

$S = \lim_{n \rightarrow \infty} S_n = \lim_{n \rightarrow \infty} a \dfrac{1-r^n}{1-r}= \frac{a}{1-r}. \nonumber$

Activity $$\PageIndex{3}$$

The formulas we have derived for the geometric series and its partial sum so far have assumed we begin indexing our sums at $$n = 0$$. If instead we have a sum that does not begin at n = 0, we can factor out common terms and use our established formulas. This process is illustrated in the examples in this activity.

a. Consider the sum

$\sum_{k=1}^\infty(2)\left(\dfrac{1}{3}\right)^k = (2)\left(\dfrac{1}{3}\right)+(2) \left(\dfrac{1}{3}\right)^2 + (2)\left(\dfrac{1}{3}\right)^3 + .... \nonumber$

Remove the common factor of (2) $$\left(\dfrac{1}{3}\right)$$ from each term and hence find the sum of the series.

b. Next let $$a$$ and $$r$$ be real numbers with $$−1 < r < 1$$. Consider the sum

$\sum_{k=3}^\infty = ar^k = ar^3 + ar^4+ ar^5 + ... \nonumber$

Remove the common factor of $$ar^3$$ from each term and find the sum of the series.

c. Finally, we consider the most general case. Let $$a$$ and $$r$$ be real numbers with $$−1 < r < 1$$, let $$n$$ be a positive integer, and consider the sum

$\sum_{k=n}^\infty = ar^k = ar^n + ar^{n+1}+ ar^{n+2} + ... \nonumber$

Remove the common factor of $$ar^n$$ from each term to find the sum of the series.

Summary

In this section, we encountered the following important ideas:

• A geometric series is an infinite sum of the form

$\sum_{k=0}^\infty ar^k \nonumber$

where $$a$$ and $$r$$ are real numbers and $$r \neq 0$$ .

• For the geometric series $$\sum_{k=0}^\infty ar^k$$, its $$n$$th partial sum is

$S_n = \sum_{k=0}^{n-1} ar^k \nonumber$

An alternate formula for the $$n$$th partial sum is

$S_n = a \dfrac{1-r^n}{1-r}. \nonumber$

Whenever |$$r$$| < 1, the infinite geometric series $$\sum_{k=0}^{\infty}ar^k$$ has the finite sum $$\dfrac{a}{ 1−r}$$.

Contributors

Matt Boelkins (Grand Valley State University), David Austin (Grand Valley State University), Steve Schlicker (Grand Valley State University)