How to Find the Value of an Infinite Sum in a Geometric Sequence
If your precalculus 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_{n}) can get. If r < 1, for every value of n, r^{n} 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 r^{k} almost disappears completely in the finite geometric sum formula:
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.