Infinite Geometric Series

Last updated
Save as PDF

Page ID: 221320

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

\( \newcommand{\dsum}{\displaystyle\sum\limits} \)

\( \newcommand{\dint}{\displaystyle\int\limits} \)

\( \newcommand{\dlim}{\displaystyle\lim\limits} \)

\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)

( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)

\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)

\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)

\( \newcommand{\Span}{\mathrm{span}}\)

\( \newcommand{\id}{\mathrm{id}}\)

\( \newcommand{\Span}{\mathrm{span}}\)

\( \newcommand{\kernel}{\mathrm{null}\,}\)

\( \newcommand{\range}{\mathrm{range}\,}\)

\( \newcommand{\RealPart}{\mathrm{Re}}\)

\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

\( \newcommand{\Argument}{\mathrm{Arg}}\)

\( \newcommand{\norm}[1]{\| #1 \|}\)

\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)

\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)

\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)

\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)

\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

\( \newcommand{\vectorC}[1]{\textbf{#1}} \)

\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)

\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)

\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

\(\newcommand{\longvect}{\overrightarrow}\)

\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)

Infinite Geometric Series

Definition of an Infinite Geometric Series

We learned that a geometric series has the form

\( \displaystyle\sum_{i=1}^{n} a_i r^{i-1} \)

Definition
The series
\( \displaystyle\sum_{i=1}^{\infty} a_i r^{i-1} \)
is called the infinite geometric series.

Calculating the Infinite Geometric Series

Example

Suppose that a runner begins on a one mile track. In the first part of the race the runner runs 1/2 of the track. In the second part of the race the runner runs half the remaining distance (1/4 miles). In the third part of the race the runner runs half the remaining distance (1/8 miles). The runner continues the process indefinitely. The runner will never complete the entire track, but after each part the runner will get closer and closer to finishing. We say that the runner will run the entire track "in the limit". The equation for this is
\( \displaystyle\sum_{i=1}^{\infty} \frac{1}{2} (\frac{1}{2})^{i-1} = 1 \)

We say that this geometric series converges to 1.

Example

Suppose that there is a geometric sequence with

        r = 2 and a₁ = 1

then the infinite geometric series is

        2 + 4 + 8 + 16 + ...

We see that these numbers just keep increasing to infinity. In general, if |r| > 1 then the geometric series is never defined. We say that the series diverges. If |r| < 1 then recall that the finite geometric series has the formula

        S _{i = 1}ⁿ a₁rⁱ^-1 = a₁[(1 - rⁿ)/(1 - r)]

If n is large then rⁿ will converge asymptotically to 0 and hence we have the formula

\( \displaystyle\sum_{i=1}^{\infty} a_1 r^{i-1} = \frac{a_1}{1-r} \)

Example:

Find
\( \displaystyle\sum_{i=1}^{\infty} 2 (\frac{1}{3})^{i-1} \)

Solution:

We have

2/(1 - 1/3) = 2/(2/3) = 3

Repeating Decimals

Recall that a rational number in decimal form is defined as a number such that the digits repeat. We can use a geometric series to find the fraction that corresponds to a repeating decimal.

Example:

        .737373737... = .73 + .0073 + .000073 + ...

we have

        a₁ = .73     and     r = .01

Hence

        .73737373.... = .73/(1 - .01) = (73/100)/(99/100) = 73/99

Example:

        4.16826826826826... = 4.1 + .1[.682 + .000682 + .000000682 + ...]

we have

        a₁ = .682 and r = .001

Hence

        4.16826826826826... = 4.1 + .1[.682/(1 - .001)

        = 41/10 + (1/10)(682/1000)/(999/1000)

        = 41/10 + 682/9990 = 41,641/9990
Interval Of Convergence

If an infinite series involves a variable x, then we call the interval of convergence the set of all x such that the interval converges for that x. For example

\( \displaystyle\sum_{i=1}^{\infty} x^{i-1} \)

has interval of convergence

        -1 < x < 1.

Example:

Find the interval of convergence of

\( \displaystyle\sum_{i=1}^{\infty} (3x-2)^{i-1} \)

Solution

We write

        |3x - 2| < 1

we solve

        3x - 2 =1 or 3x - 2 = -1

Adding 2 and dividing by three gives

        x = 1     or     x = 1/3

Hence

        1/3 < x < 1

is the interval of convergence.

Back to the Sequences and Series Site

Search

Text Color

Text Size

Margin Size

Font Type