The Geometric Series


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

displaymath223

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

displaymath224


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

displaymath232

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

displaymath233

multiply both sides by q

displaymath234

then subtract the second line from the first:

displaymath235

For tex2html_wrap_inline258 , we can solve this for tex2html_wrap_inline260 :

displaymath236

It is not hard to see what happens when we consider

displaymath237


Summarizing:

The identity

displaymath238

is valid exactly when -1<q<1.


Try it yourself!

Find the sum of the series

displaymath276

Click here for the answer or to continue.


Helmut Knaust

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