[BruceZ]

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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.

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[1 - C^(N+1)] / (1 - C) is the sequence of partial sums which I mentioned. An infinite series is defined as the limit of a sequence of partial sums. This sequence of partial sums has no limit for |C| > 1, hence the series diverges by definition.

This definition brings more clarity to the elementary notion that an infinite series is something of the form x1 + x2 + x3 + .... as it makes clear what the "..." means, and helps prevent us from trying to sum series which don't converge. A similar notion allows us to make sense of such things as continued fractions:

1+1/(1+1/(1+1/(1+...

and continued square roots:

sqrt(1+sqrt(1+sqrt(1 + ...

BTW, both of these equal the golden ratio 1/2 + sqrt(5)/2.