Does ∑n=1∞1n2+1\sum_{n=1}^{\infty} \frac{1}{n^2+1}∑n=1∞n2+11 converge?
Yes, by comparison with ∑1n2\sum \frac{1}{n^2}∑n21.
No, it is harmonic.
Yes, by the Divergence Test.
No, it grows.