The Geometric Series

The easiest (but not the only) way is to factor out

The series inside the parentheses is the familiar geometric series with
. Thus, this series sums to

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For which values of q will this work?

The summation trick on the previous page does not work for all values of q. Consider for instance q=1. Clearly, the sum

does not add up to a finite number! One says that this series diverges (= is not convergent). This does not have much to do with the fact that in the end we "divide by 0"; try q=2 or q=-1.

The problem lies much deeper. The sad truth is that many of the algebraic properties of finite sums do not work for infinite sums--troubling mathematicians over the centuries! So let's be very cautious and try again. This time we only consider finite sums and then take the limit! Let

multiply both sides by q

then subtract the second line from the first:

For , we can solve this for :

It is not hard to see what happens when we consider

For q>1, the expressions go to infinity, so there is no limit.
For q<-1, the expressions alternate between big positive and big negative numbers, so there is no limit.
For q=-1, the expressions alternate between -1 and 1, so there is no limit.
For -1<q<1, the expressions tend to zero; so tends to .

Summarizing:

The identity

is valid exactly when -1<q<1.

Try it yourself!

Find the sum of the series

Click here for the answer or to continue.

Helmut Knaust