Find values of $p$ for which the series is convergent.

Question

I am still having a hard time understanding how to find what values of $p$ allow for the series to converge.

Here is the series I am currently trying to find $p$ for: $$\sum_{n=1}^\infty n(1+n^2)^p$$

I tried doing an integral test with $p \gt 0$ but I got a wrong answer I believe.

Thanks.

Answer 1

By the integral test consider $\int_{x=1}^{\infty}x(1+x^2)^pdx$. By substitution with $u = 1+x^2$, this is equal to $\frac{1}{2}\int_{u=2}^{\infty}u^pdu$.

$\frac{1}{2}\int_{u=2}^{\infty}u^pdu$ is known to converge $\forall p<-1$, thus the series you provided will converge for all $p<-1$.

Answer 2

Since the harmonic series is divergent, p must be lesser than $-1$.