The Geometric Series

The easiest (but not the only) way is to factor out tex2html_wrap_inline228 :


The series inside the parentheses is the familiar geometric series with
tex2html_wrap_inline230. Thus, this series sums to


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 tex2html_wrap_inline258 , we can solve this for tex2html_wrap_inline260 :


It is not hard to see what happens when we consider



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

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