I know the taylor series of $\log(1+x)$. However I don't understand how to find the convergence for $x>1$ and divergence if $x<1$.

  • 3
    $\begingroup$ Hint, integrate the alternating geometric series $\endgroup$ – imranfat Dec 6 '13 at 0:06
  • $\begingroup$ Alternately, differentiate n times. The ratio test will give you a radius of convergence of 1. $\endgroup$ – Jean-Claude Arbaut Dec 6 '13 at 0:29
  • $\begingroup$ I'm voting to close this question as off-topic because it is about two years old with, by now, a known solution $\endgroup$ – Leucippus Jul 16 '15 at 3:31

We have (i) convergence if $|x|\lt 1$, and divergence if $|x|\gt 1$. This can be done by using the Ratio Test.

We also have (ii) convergence at $x=1$ and divergence at $x=-1$. For $x=1$, we have an alternating series. For $x=-1$, we get a close relative of a familiar divergent series.


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