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Math Help - proving a series converges

  1. #1
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    proving a series converges

    i have obtained the series for In(1+x) how can i find the radius of convergence? i have tried the ratio test but this doesn't seem to work for [(-1)^k* x^(k+1)]/k+1
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  2. #2
    MHF Contributor Also sprach Zarathustra's Avatar
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    ln(1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}-...=\sum^{\infty}_{n=1} \frac{(-1)^{n+1}} {n}x^n


    Let us look on (*) \sum^{\infty}_{n=1}| \frac{(-1)^{n+1}}{n}x^n|=\sum^{\infty}_{n=1} \frac{1}{n}|x^n|

    Using Cauchy test, (*) converges if lim_{n\to\infty}\sqrt[n]{ \frac{1}{n}|x^n|}<1

    Or:

    |x|<1

    Now, check what happens when x=1 and x=-1.
    Last edited by Also sprach Zarathustra; March 19th 2011 at 11:17 PM.
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