# Infinite Geometric Series

How to solve the infinite geometric sequence problems: formula, proof, examples, and their solutions.

## Formula

The infinite geometric series means*S* = *a*_{1} + *a*_{2} + *a*_{3} + ...

= *a*_{1} + *a*_{1}*r* + *a*_{1}*r*^{2} + ... .

Geometric series

If -1 < *r* < 1,*S* gets close to a constant number.*S* = *a*_{1} / (1 - *r*)*S*: infinite sum*a*_{1}: first term*r*: common ratio

## Proof

Start from *S*_{n}:*S*_{n} = *a*_{1}(1 - *r*^{n}) / (1 - *r*)

Geometric series

If -1 < *r* < 1 and *n* → ∞,

then *r*^{n} becomes 0.

Think of (1/2)^{n}: 1/2, 1/4, 1/8, 1/16, ... .

Then the infinite sum, *S*, becomes like this:*S* = *a*_{1}(1 - 0) / (1 - *r*).

So *S* = *a*_{1} / (1 - *r*).

## Example 1

*a*_{1} = 1*r* = 1/2(-1 < 1/2 < 1)*S* = 1 / (1 - 1/2)

Multiply 2 to both of the numerator and the denominator.

## Example 2

*n* goes from 1 to ∞.

So use the infinite geometric series formula.*a*_{1} = 6/7*r* = 1/7(-1 < 1/7 < 1)*S* = (6/7) / (1 - 1/7)