Re: A Stupid(?) Question
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Sum of Geometric Series converges when C < 1. One of the proofs of this that I learned back in school follows.
So, S = C^0 + C^1 + C^2 + ...
C*S = C^1 + C^2 + C^3 + ...
S - C*S = C^0
S = C^0 / (1 - C) = 1 / (1 - C)
Now, comes the stupid question... how does this proof fail for C > 1? What is the reason why this proof is valid when C < 1 and invalid when C > 1? At C = 1, we clearly see that the answer is undefined... but at C > 1, S is equal to some negative number, which is obviously wrong. But, my question, lame as it may be, is exactly where is it that this proof becomes invalid?
Thanks.
-RMJ
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When determining the sum of a series, it is necessary to show that the series converges, not just what the series would be equal to if it did converge. For C > 1, it is easy to show that the series does not converge, or more precisely, that the sequence of partial sums does not converge to a limit. S - C*S = C^0 doesn't make any sense when S and C*S are infinite.
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