# determine the convergence of this infinity serie [closed]

determine the convergence of $$\sum^{\infty}_{n=1}n^{\frac{p^2 -2}{p^2 +2p -3}}$$ how do I prove this?

• Do you know when $\sum_{n=1}^\infty n^c$ converges? Feb 9 '21 at 18:26

To avoid confusion, I will call your $$p$$ in my answer $$r$$.
The series $$\sum_{n=1}^{\infty}n^{-p}$$ is called $$p$$-series and known to be convergent iff $$p>1$$. Thus, your series is convergent if and only if $$-\frac{r^2-2}{r^2+2r-3}>1$$, which holds for all $$r \in (-3,\frac{-1-\sqrt{11}}{2}) \cup (1,\frac{-1+\sqrt{11}}{2})$$.
If you are interested in $$p$$-series, you should also take a look at the Riemann zeta function, which is the extension to complex $$s$$ instead of $$p$$.