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When does this series converge?$$ \sum_{n=1}^\infty \frac{a^n}{n^b} $$ I want to know the condition of a and b.

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aside: this is a polylogarithm $\mathrm{Li}_{b}(a)$ ( – deoxygerbe Jun 9 '12 at 4:21
up vote 2 down vote accepted

Hint: I assume we are working over the reals.

For $|a|\lt 1$, use Ratio Test.

For $|a|\gt 1$, terms don't go to $0$, or Ratio Test.

This leaves $a=1$ and $a=-1$.

For $a=1$, comparison with the harmonic series if $b \le 1$, and Integral Test for $b \gt 1$.

For $a=-1$, terms don't go to $0$ if $b\le 0\,$, alternating series if $b\gt 0$.

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Typo in last line, $b=-1$ should read $a=-1$. – Erick Wong Jun 8 '12 at 21:38
@ErickWong: Thank you, fixed! – André Nicolas Jun 8 '12 at 22:59

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