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I was doing a bit of math when I came across logarithmic series. I have no idea from where they come from. They seem so unrelated, that I have no intuition behind them at all.
So, can anyone prove that $\displaystyle\sum_{n=1}^{\infty}\frac{(-1)^{n+1}x^{n}}{n}=\ln(1+x)$ $\forall -1<x\leq1$
or atleast point me in the right direction?

AvZ
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  • Taylor Series of $\ln (1+x)$. – r9m Jan 10 '15 at 17:37
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    You are probably familiar with the geometric series $1-t+t^2-t^3+t^4-t^5+\cdots$, which has sum $\frac{1}{1+t}$ if $|t|\lt 1$. Integrate this series term by term. – André Nicolas Jan 10 '15 at 17:37
  • @AndréNicolas, Yeah, I think I got it now. You should put it as an answer so I can accept it. – AvZ Jan 10 '15 at 17:40
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    The problem is that I am pretty sure this is a duplicate. Also, in principle one should show that integrating term by term is valid, that is, $\int_0^x \frac{1}{1+t},dt$ is equal to the result of the term by term integration (it is). – André Nicolas Jan 10 '15 at 17:44
  • @AndréNicolas, I searched for this question, but surprisingly didn't find anything. About the integration, I think Newton and Leibniz did that already for us... :) – AvZ Jan 10 '15 at 17:48

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