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If in the series $1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}\cdots$ the order of the terms be altered, so that the ratio of the number of positive terms to the number of negative terms in the first $n$ terms is ultimately $a^2$, show that the sum of the series will become $\log 2a$.

It is easy to show that when $a^2=1$, the sum is just the expansion of $\log(1+x)$ when $x=1$. However I don't know how to prove it in the general case.

Michael Hardy
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an offer can't refuse
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  • It is worth pointing out that the alternating harmonic series is on the boundary of the interval of convergence of the Maclaurin series for $\log(1+x)$. Although it can be easily shown to converge with the Alternating Series Test, showing that its limit is $\log(2)$ is something slightly less straightforward. – Theo Bendit Jul 27 '15 at 04:46
  • @TheoBendit The problem is duplicate, however the accepted answer is hard to understand. It jumps too much: "Usually, if you take the altered harmonic series and sum p positive terms and then add q negative terms, you'll get:" I don't know how to get this. It statethe answer directly without enough prove. Can you prove that answer? – an offer can't refuse Jul 27 '15 at 05:57
  • Can you identify the statement(s) in the accepted answer you specifically don't understand? That is likely enough to warrant a separate question. Also, consider asking clarifying questions as comments. –  Jul 27 '15 at 06:01

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