When does this series converge?$ \sum_{n=1}^\infty \frac{a^n}{n^b} $ I want to know the condition of a and b.
Comparison Test about the series $ \sum_{n=1}^\infty \frac{a^n}{n^b} $
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calculus
real-analysis
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0aside: this is a polylogarithm $\mathrm{Li}_{b}(a)$ (http://en.wikipedia.org/wiki/Polylogarithm) – 2012-06-09
1 Answers
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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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0@ErickWong: Thank you, fixed! – 2012-06-08