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

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    $\begingroup$ Do you know when $\sum_{n=1}^\infty n^c$ converges? $\endgroup$ Feb 9 '21 at 18:26
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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 $r \in (-3,\frac{-1-\sqrt{11}}{2}) \cup (1,\frac{-1+\sqrt{11}}{2})$.

Further research
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 $p$.

EDIT: Does that answer your question?

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