How to Solve Infinite Geometric Series

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: \(54, \ 18, \ 6, \ 2, \ \frac{2}{3}, \ ...\)

 

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: \(54, \ 18, \ 6, \ 2, \ \frac{2}{3}, \ ...\)

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

Infinite Geometric Series Practice Quiz