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Sums of Infinite Geometric Series

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Finding the Sum of an Infinite Geometric Series

Your task as Agent Infinite Geometric Series, should you choose to accept it, is to find the sum of the geometric series \sum \limits_{n=1}^{\infty}3 \left(\frac{1}{3}\right)^{n-1} .

(NOTE:  The large E looking letter with the infinity symbol above simply indicates that we are to find the INFINITE sum of the geometric sequence.  Don't let it intimidate you!)

Guidance

In the previous concept we explored partial sums of various infinite series and observed their behavior as n became large to see if the sum of the infinite series was finite. Now we will focus our attention on geometric series. Look at the partial sums of the infinite geometric series below:

Series \sum \limits_{n=1}^{\infty}3(1)^{n-1} \sum \limits_{n=1}^{\infty}10 \left(\frac{3}{4}\right)^{n-1} \sum \limits_{n=1}^{\infty}5 \left(\frac{6}{5}\right)^{n-1} \sum \limits_{n=1}^{\infty}(-2)^{n-1} \sum \limits_{n=1}^{\infty}2 \left(- \frac{1}{3}\right)^{n-1}
S_5 15 30.508 37.208 11 1.506
S_{10} 30 37.747 129.793 -341 1.5
S_{50} 150 40 227485.954 -3.753 \times 10^{14} 1.5
S_{100} 300 40 2070449338 -4.226 \times 10^{29} 1.5

From the table above, we can see that the two infinite geometric series which have a finite sum are \sum \limits_{n=1}^{\infty}10 \left(\frac{3}{4}\right)^{n-1} and \sum \limits_{n=1}^{\infty}2 \left(- \frac{1}{3}\right)^{n-1} . The two series both have a common ratio, r , such that \left | r \right \vert < 1 or -1<r<1 .

Take a look at the formula for the sum of a finite geometric series: S_n=\frac{a_1\left(1-r^n\right)}{1-r} . What happens to r^n if we let n get very large for an r such that \left |r \right \vert <1 ? Let’s take a look at some examples.

r values r^5 r^{25} r^{50} \ldots r^n or r^{\infty}
\frac{5}{6} 0.40188 0.01048 0.00011 0
- \frac{4}{5} -0.32768 -0.00378 0.00001 0
1.1 1.61051 10.83471 117.39085 keeps growing
- \frac{1}{3} -0.00412 -1.18024 \times 10^{-12} 1.39296 \times 10^{-24} 0

This table shows that when \left | r \right \vert <1 , r^n=0 , for large values of n . Therefore, for the sum of an infinite geometric series in which \left | r \right \vert <1, S_{\infty}=\frac{a_1 \left(1-r^n\right)}{1-r}=\frac{a_1\left(1-0\right)}{1-r}=\frac{a_1}{1-r} .

Example A

Find the sum of the geometric series if possible. \sum \limits_{n=1}^{\infty}100 \left(\frac{8}{9}\right)^{n-1} .

Solution: Using the formula with a_1=100 , r=\frac{8}{9} , we get S_{\infty}=\frac{100}{1- \frac{8}{9}}=\frac{100}{\frac{1}{9}}=900 .

Example B

Find the sum of the geometric series if possible. \sum \limits_{n=1}^{\infty}9 \left(\frac{4}{3}\right)^{n-1} .

Solution: In this case, \left | r \right \vert=\frac{4}{3}>1 , therefore the sum is infinite and cannot be determined.

Example C

Find the sum of the geometric series if possible. \sum \limits_{n=1}^{\infty}5(0.99)^{n-1}

Solution: In this case a_1=5 and r=0.99 , so S_{\infty}=\frac{5}{1-0.99}=\frac{5}{0.01}=500 .

Intro Problem Revisit In this case a_1=3 and r=\frac{1}{3} , so S_{\infty}=\frac{3}{1-\frac{1}{3}}=\frac{3}{\frac{2}{3}}=\frac{9}{2}=4.5 .

Guided Practice

Find the sums of the following infinite geometric series, if possible.

1. \sum \limits_{n=1}^{\infty}\frac{1}{9} \left(- \frac{3}{2}\right)^{n-1}

2. \sum \limits_{n=1}^{\infty}4 \left(\frac{7}{8}\right)^{n-1}

3. \sum \limits_{n=1}^{\infty}3(-1)^{n-1}

Answers

1. \left | r \right \vert=\left |- \frac{3}{2} \right \vert=\frac{3}{2}>1 so the infinite sum does not exist.

2. a_1=4 and r=\frac{7}{8} so S_{\infty}=\frac{4}{1- \frac{7}{8}}=\frac{4}{\frac{1}{8}}=32 .

3. \left | r \right \vert=\left |-1 \right \vert=1 \ge 1 , therefore the infinite sum does not converge. If we observe the behavior of the first few partial sums we can see that they oscillate between 0 and 3.

S_1&=3 \\S_2&=0 \\S_3&=3 \\S_4&=0

This pattern will continue so there is no determinable sum for the infinite series.

Practice

Find the sums of the infinite geometric series, if possible.

  1. \sum \limits_{n=1}^{\infty} 5 \left(\frac{2}{3}\right)^{n-1}
  2. \sum \limits_{n=1}^{\infty} \frac{1}{10} \left(- \frac{4}{3}\right)^{n-1}
  3. \sum \limits_{n=1}^{\infty} 2 \left(- \frac{1}{3}\right)^{n-1}
  4. \sum \limits_{n=1}^{\infty} 8(1.1)^{n-1}
  5. \sum \limits_{n=1}^{\infty} 6(0.4)^{n-1}
  6. \sum \limits_{n=1}^{\infty} \frac{1}{2} \left(\frac{3}{7}\right)^{n-1}
  7. \sum \limits_{n=1}^{\infty} \frac{5}{3} \left(\frac{1}{6}\right)^{n-1}
  8. \sum \limits_{n=1}^{\infty} \frac{1}{5} (1.05)^{n-1}
  9. \sum \limits_{n=1}^{\infty} \frac{4}{7} \left(\frac{6}{7}\right)^{n-1}
  10. \sum \limits_{n=1}^{\infty} 15 \left(\frac{11}{12}\right)^{n-1}
  11. \sum \limits_{n=1}^{\infty} 0.01 \left(\frac{3}{2}\right)^{n-1}
  12. \sum \limits_{n=1}^{\infty} 100 \left(\frac{1}{5}\right)^{n-1}
  13. \sum \limits_{n=1}^{\infty} \frac{1}{2} \left(\frac{5}{4}\right)^{n-1}
  14. \sum \limits_{n=1}^{\infty} 2.5(0.85)^{n-1}
  15. \sum \limits_{n=1}^{\infty} -3 \left(\frac{9}{16}\right)^{n-1}

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