[AaronBrown]

This is an excellent question. BruceZ's answer is correct, in general you can have a set of perfectly legal steps that leads to an incorrect result if there in an infinity mixed in. But this case also has a more tangible answer. Do the finite case:

S = C^0 + C^1 + C^2 + ... + C^N
C*S = C^1 + C^2 + C^3 + ... + C^(N+1)
S - C*S = [C^0 - C^(N+1)]
S = [1 - C^(N+1)] / (1 - C)

If |C|<1 then C^(N+1) goes to zero as N goes to infinity. So the limit of the finite sum equals 1/(1 - C). Another way of saying that is the series converges to 1/(1 - C). But if |C| > 1 then the sum diverges, there is no finite limit.