# Infinite Geometric Series

A geometric sequence in which the number of terms increases without bound is called an **infinite geometric series**.

If the absolute value of the common ratio is less than , , the sum of terms, always approaches a definite limit as increases without bound. As we have proved that the sum of a finite geometric series is

We rewrite it as

If now is numerically less than , i.e., , the numerical value of decreases as increases and by taking sufficiently large, we can make as small as we want. Hence, by taking large enough, we can make differ from by as little as we want, i.e., we can make approach as a limit. Symbolically

Where is the sum of an infinite geometric progression with first term as and common ratio . We can also call it as **Infinite Series**. According, the expression is called an **infinite geometric series**. If the terms continuous decease as approaches a limiting value as becomes infinitely large, it is said to be a **Convergent Infinite Series**.

__Example__:

Find the sum of the infinite geometric sequence

__Solution__:

We have

, , then

__Recurring or Periodic Decimals__:

An interesting application of a geometric progression with infinitely many terms in evaluation of **recurring or periodic decimals**.

When we attempt to express a common fraction such as or as a decimal fraction, the decimal always either terminates or ultimately repeats in blocks. Thus

(Decimal Terminates)

(Decimal Repeats)

In the division process by which we express the fraction as a decimal fraction the reminders can only be the numbers . If at any stage in the division we obtain a remainder , the process terminates. Otherwise, after not more than divisors, one of the remainders must recur and the decimal begins to repeat.

__Example__:

Express the recurring decimal fraction as a common fraction.

__Solution__:

Given decimal fraction can be written in the form

Hence our number consists of the decimal plus the sum of an infinite geometric progression with first term and common ratio . The sum of the infinite progression is expressible as the fraction

Hence