Sum of Infinite Geometric Series

In this week’s Math 11 Pre-Cal I learned that you can actually determine the exact Sum of Infinite Geometric Series in certain situations, when they are converges.( when -1<r<1) as with a rate less than 1 and greater than -1 the next term will get closer and closer to zero, therefore there is a determinable  sum. In the case of a diverging series, the  sum will get infinity big and therefor we can’t determine the exact sum.

The equation for the sum of an regular Geometric Series is

Sn=$\frac{a(1-r^n)}{1-r}$

When -1<r<1 $r^n$ approaches 0 as n increases indefinitely.

So, Sn approaches Sn= $\frac{a(1-0)}{1-r}$, therefor $S\infty= \frac{a}{1-r}$

EX:

A Infinite Geometric Series where r=0.5

8,4,2,1

$S\infty= \frac{8}{1-0.5}$ $S\infty= \frac{8}{0.5}$ $S\infty= {16}$