Expert: Steve Holleran - 5/22/2009

Question
Hi, I have a math test tomorrow and there is a question on my review sheet
that I just can't figure out, here's the question

For the geometric series, find the sum.  If the series has no sum, say so.
1/2-1/3+2/9-4/27+..

If you could explain how to do it, that would be great!
Thank you,
Jess

Answer
Hi Jess,

Okay, an infinite geometric series will have a sum if the ratio (r) between successive terms is between -1 and 1 (another way of saying this is if its absolute value is between 0 and 1).

Here, you can find the ratio by dividing any two neighboring terms.
Lets say you took the second and third ones:

           (2/9) / (-1/3) = (-2/3) = r

so, this series has a sum.

Now, the formula for the sum of an infinite geometric series is

                  S =  a / (1 - r)   where a is the first term
                                     and r is the common ratio.

So, here we have:

                  S = (1/2) / (1 - (-2/3))

                    = (1/2) / (5/3)  = 3/10

And this would be the sum of the series.

By the way, if the ratio is ever more than 1 or less than -1, then the infinite series has no sum (and is called divergent).

Hope this helps in time.
Steve