I have to find whether these series converge.

$\displaystyle \sum {\frac {n+1} {n^2 + 1}}$ by using the limit comparison test, and using $\displaystyle \sum {b_n}$ = $\displaystyle \frac {1} {n}$ i found that it diverges.

As for this example: $\displaystyle \sum {\frac {n!} {n^n}}$ i used the ratio test and found it diverges also.

Can someone check if i am correct, and also help me with this one

$\displaystyle \sum {\frac {1} {2^n - 1 + cos^2(n^3)}}$