The series obtained by adding the terms of a G.P. is called the **geometric series**. Let be the sum of the first terms of the G.P. The term is , the term is , etc. Hence

--- (1)

Multiplying both members of (1) by , we get

--- (2)

On subtracting each side of (2) from the corresponding side of (1), we obtain

If the common ratio in a G.P. is more than , then each successive term is greater than the previous one and the sum of the terms grows very rapidly and tends to an infinity as tends to infinity.

If the common ratio is less than , then each term will be smaller than the previous one and the total will increase but will not exceed a finite value as tends to an infinity.

Thus

__Example__:

Using G.P., find the sum of .

__Solution__:

** ** We have

, and

Substituting these values in the given formula, we get

(as )

__Example__:

Given , , , find .

__Solution__:

** **Since,