A geometric sequence in which the number of terms increases without bounds 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 bounds. As we have proved, 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. By taking which is 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
Here is the sum of an infinite geometric progression with the first term as and common ratio . We can also call it an infinite series. Accordingly, the expression is called an infinite geometric series. If the terms continuously decrease as approaches a limiting value as becomes infinitely large, it is said to be a convergent infinite series.
Find the sum of the infinite geometric sequence
, , then
Recurring or Periodic Decimals
An interesting application of a geometric progression with infinitely many terms is the 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
In the division process by which we express the fraction as a decimal fraction the remainders can only be the numbers . If at any stage in the division we obtain a remainder of , the process terminates. Otherwise, after not more than divisors, one of the remainders must recur and the decimal begins to repeat.
Express the recurring decimal fraction as a common fraction.
The 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 the first term and common ratio . The sum of the infinite progression is expressible as the fraction