How to Solve Infinite Geometric Series

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Infinite Geometric Series

A geometric series with an infinite number of terms is called an infinite geometric series. The infinite geometric series is shown as $a, \ ar, \ ar^2, \ ar^3, \ ... \ ,$ to $∞$ .

Infinite Geometric Formula

If the value of $r$ is between $1$ and $-1$ , then the formula for an infinite series is: $S_n \ = \ \frac{a}{1 \ - \ r}$
Where,

$a \ =$ The first term
$r \ =$ The common ratio

Only if $r$ is less than one does the geometric series add up to a sum. If $r$ is greater than one, the series doesn't converge, and there is no sum.

Example

The geometric series is $2, \ 8, \ 32, \ 128, \ ...$

To find the sum of $2, \ 8, \ 32, \ 128, \ ...,$ first we need to find the value of $r$ : $r \ = \ \frac{8}{2} \ = \ 4$

Since $r \ > \ 1$ , the series does not converge and has no sum.

Free printable Worksheets

Exercises for Infinite Geometric Series

1) Determine whether the geometric series converges or diverges: $4, \ 12, \ 36, \ 108, \ 324, \ ...$

2) Determine whether the geometric series converges or diverges: $-7, \ 14, \ -28, \ 56, \ -112, \ ...$

3) Determine whether the geometric series converges or diverges: $36, \ 18, \ 9, \ \frac{9}{2} \ , \ \frac{9}{4} \ , \ ...$

4) Determine whether the geometric series converges or diverges: $486, \ -162, \ 54, \ -18, \ 6, \ ...$

5) Determine whether the geometric series converges or diverges: $\frac{3}{64} \ , \ \frac{3}{16} \ , \ \frac{3}{4} \ , \ 3, \ 12, \ ...$

6) Determine whether the geometric series converges or diverges: $\frac{7}{8} \ , \ \frac{7}{16} \ , \ \frac{7}{32} \ , \ \frac{7}{64} \ , \ \frac{7}{128} \ , \ ...$

7) Evaluate the geometric series described: $5, \ 20, \ 80, \ 320, \ 1280, \ ...$

8) Evaluate the infinite geometric series described: $1024, \ 256, \ 64, \ 16, \ 4, \ ...$

9) Evaluate the geometric series described: $336, \ 168, \ 84, \ 42, \ 21, \ ...$

10) Evaluate the geometric series described: $54, \ 18, \ 6, \ 2, \ \frac{2}{3}, \ ...$

Answers

1) Determine whether the geometric series converges or diverges: $4, \ 12, \ 36, \ 108, \ 324, \ ...$

$\color{red}{r \ = \ \frac{12}{4} \ = \ 3 \ ⇒ \ diverges}$

2) Determine whether the geometric series converges or diverges: $-7, \ 14, \ -28, \ 56, \ -112, \ ...$

$\color{red}{r \ = \ \frac{14}{-7} \ = \ -2 \ ⇒ \ diverges}$

3) Determine whether the geometric series converges or diverges: $36, \ 18, \ 9, \ \frac{9}{2} \ , \ \frac{9}{4} \ , \ ...$

$\color{red}{r \ = \ \frac{18}{36} \ = \ \frac{1}{2} \ ⇒ \ converges}$

4) Determine whether the geometric series converges or diverges: $486, \ -162, \ 54, \ -18, \ 6, \ ...$

$\color{red}{r \ = \ \frac{-162}{486} \ = \ -\frac{1}{3} \ ⇒ \ converges}$

5) Determine whether the geometric series converges or diverges: $\frac{3}{64} \ , \ \frac{3}{16} \ , \ \frac{3}{4} \ , \ 3, \ 12, \ ...$

$\color{red}{r \ = \ \frac{\frac{3}{16}}{\frac{3}{64}} \ = \ 4 \ ⇒ \ diverges}$

6) Determine whether the geometric series converges or diverges: $\frac{7}{8} \ , \ \frac{7}{16} \ , \ \frac{7}{32} \ , \ \frac{7}{64} \ , \ \frac{7}{128} \ , \ ...$

$\color{red}{r \ = \ \frac{\frac{7}{16}}{\frac{7}{8}} \ = \ \frac{1}{2} \ ⇒ \ converges}$

7) Evaluate the geometric series described: $5, \ 20, \ 80, \ 320, \ 1280, \ ...$

$\color{red}{r \ = \ \frac{20}{5} \ = \ 4 \ ⇒ \ r \ > \ 1 \ ⇒ \ diverges \ ⇒ \ No \ sum}$

8) Evaluate the infinite geometric series described: $1024, \ 256, \ 64, \ 16, \ 4, \ ...$

$\color{red}{r \ = \ \frac{256}{1024} \ = \ \frac{1}{4}}$
$\color{red}{S_n \ = \ \frac{a}{1 \ - \ r} \ = \ \frac{1024}{1 \ - \ \frac{1}{4}} \ = \ \frac{1024}{-\frac{3}{4}} \ = \ -\frac{4096}{3}}$

9) Evaluate the geometric series described: $336, \ 168, \ 84, \ 42, \ 21, \ ...$

$\color{red}{r \ = \ \frac{168}{336} \ = \ \frac{1}{2}}$
$\color{red}{S_n \ = \ \frac{a}{1 \ - \ r} \ = \ \frac{336}{1 \ - \ \frac{1}{2}} \ = \ 672}$

10) Evaluate the geometric series described: $54, \ 18, \ 6, \ 2, \ \frac{2}{3}, \ ...$

$\color{red}{r \ = \ \frac{18}{54} \ = \ \frac{1}{3}}$
$\color{red}{S_n \ = \ \frac{a(r^n \ - \ 1)}{r \ - \ 1} \ = \ \frac{54}{1 \ - \ \frac{1}{3}} \ = \ 81}$

How to Solve Infinite Geometric Series