Choose for what constants the series converges or diverges I have a series
$$
\sum_{n=1}^\infty \frac{1}{a^n + 1}
$$
where $a > 0$.
I need to find for which values $a$ the series is convergent and divergent.
I have no idea what to do.
I don't think I can apply neither ratio test, integral test, comparison tests.
I'm unsure if I can use alternating series test since $a^n$ potentially could alternate; however, I have the condition $a > n$, so that doesn't apply neither.
I know that it is easier to decide if I first look at $0 < a \leq 1$ and then for $a > 1$.

If $a=1$ then the general term equals $\frac{1}{2} \nrightarrow 0,\;\;n\to \infty,$ hence the series is divergent.
If $0<a<1$ then $0<a^n<1 \Rightarrow 1<a^n +1<2 \Rightarrow \frac{1}{2}<\frac{1}{a^n +1}<1.$ In this case the necessary condition for convergence does not hold too.
If $a>1$ then $\frac{1}{a^n +1}<\frac{1}{a^n}$ and the series is convergent by ratio test.
