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A Stupid(?) Question
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 |
#2
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Re: A Stupid(?) Question
[ 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. |
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Re: A Stupid(?) Question
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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Re: A Stupid(?) Question
[ QUOTE ]
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. [/ QUOTE ] [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. |
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