Comparison Test about the series $ \sum_{n=1}^\infty \frac{a^n}{n^b} $

Question

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

Answer 1

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 $b\gt 0$.