The series inside the parentheses is the familiar geometric series with
. Thus, this series sums to
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
is valid exactly when -1<q<1.
Find the sum of the series
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