The Geometric Series

The Geometric Series - The Geometric Series

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http://www.sosmath.com/calculus/geoser/geoser02.html 1 of 2 1/31/08 8:56 AM 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 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
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This note was uploaded on 07/16/2009 for the course MATH 3705 taught by Professor Jaberabdualrahman during the Winter '08 term at Carleton CA.

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The Geometric Series - The Geometric Series

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