I first break it up. (n+1)^2=n^2+2n+1
Now you can evaluate three separate Summations.
I think the 2nd one can be evaluate like a geometric since ArcTan(Infinity)=pi/2
Do these converge absolutely, conditionally, or diverge?
sigma from n = 1 to infinity of (n + 1)^2 / n!
sigma from n = 1 to infinity of (1/ (arctan(n))^(2n)
so for the first one I applied the ratio test and got
lim (as n goes to infinity) of (n^2 + 2n + 1) / ( (n+1)(n^2) ) + lim (as n goes to infinity) (2n + 1)/( (n+1)(n^2) + lim (as n goes to infinity) of 1/(n+1)
and since all of these go to zero then the sum converges absolutely?
and for the second one you have
sum (from 1 to infinity) 1 / (pi/2)^2n
and because pi/2 is greater than 1 it just diverges by geometric series test?