A Stupid(?) Question

[BruceZ]

[ QUOTE ]
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

[/ QUOTE ]

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.