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 ratiois 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 makeas 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 asand 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