## How to Solve Infinite Geometric Series

### 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.

### 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: $$54, \ 18, \ 6, \ 2, \ \frac{2}{3}, \ ...$$

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

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