Does the series ∑n=1∞n+1n2\sum_{n=1}^{\infty} \frac{n+1}{n^2}∑n=1∞n2n+1 converge?
Yes, by the ppp-series test.
No, it behaves like ∑1n\sum \frac{1}{n}∑n1.
Yes, by the Ratio Test.
No, by the Divergence Test.