# Geometric Series

A sequence of numbers {*a*_{n}} is called a geometric sequence if the quotient of successive terms is a constant, called the common ratio. Thus *a*_{n+1}/*a*_{n} = *q* or *a*_{n+1} = *qa*_{n} for all terms of the sequence. It's supposed that *q* ≠ 0 and *q* ≠ 1.

For any geometric sequence:

A geometric series is the indicated sum of the terms of a geometric sequence. For a geometric series with *q* ≠ 1,

We say that the geometric series converges if the limit \(\lim\limits_{n \to \infty } {S_n}\) exists and is finite. Otherwise the series is said to diverge.

Let

be a geometric series. Then the series converges to \(\frac{{{a_1}}}{{1 - q}}\) if \(\left| q \right| \lt 1,\) and the series diverges if \(\left| q \right| \gt 1.\)

## Solved Problems

### Example 1.

Find the sum of the first \(8\) terms of the geometric sequence

Solution.

Here \({a_1} = 3\) and \(q = 2.\) For \(n = 8\) we have

### Example 2.

Find the sum of the series

Solution.

This is an infinite geometric series with ratio \(q = -0,37.\) Hence, the series converges to