# How to Find the Value of an Infinite Sum in a Geometric Sequence

If your pre-calculus teacher asks you to find the value of an infinite sum in a geometric sequence, the process is actually quite simple — as long as you keep your fractions and decimals straight. If *r* lies outside the range –1 __<__ *r* __<__ 1, *a*_{n}* *grows without bound infinitely, so there's no limit on how large the absolute value of *a** _{n}* (|

*a*

*|) can get. If |*

_{n}*r*| < 1, for every value of

*n,*|

*r*

*| continues to decrease infinitely until it becomes arbitrarily close to 0. This decrease is because when you multiply a fraction between –1 and 1 by itself, the absolute value of that fraction continues to get smaller until it becomes so small that you hardly notice it. Therefore, the term*

^{n}*r*

*almost disappears completely in the finite geometric sum formula:*

^{k}And if the *r*^{k}* *disappears — or gets very small — the finite formula changes to the following and allows you to find the sum of an infinite geometric series:

For example, follow the steps to find this value:

Find the value of

*a*_{1}by plugging in 1 for*n.*Calculate

*a*_{2}by plugging in 2 for*n.*Determine

*r*.To find

*r,*you divide*a*_{2}by*a*_{1}*:*Plug

*a*_{1}and*r*into the formula to find the infinite sum.Plug in and simplify to find the following:

Repeating decimals also can be expressed as infinite sums. Consider the number 0.5555555. . . . You can write this number as 0.5 + 0.05 + 0.005 + . . . , and so on forever. The first term of this sequence is 0.5; to find *r,* 0.05 divided by 0.5 = 0.1.

Plug these values into the infinite sum formula:

Keep in mind that this sum is finite only if *r* lies strictly between –1 and 1.