Calculus of a Single Variable, Chapter 9, 9.2, Section 9.2, Problem 11

sum_(n=1)^oo n^2/(n^2+1)
To verify if the series diverges, apply the nth-Term Test for Divergence.
It states that if the limit of a_n is not zero, or does not exist, then the sum diverges.

lim_(n->oo) a_n != 0 or lim_(n->oo) = DNE
:. sum a_n diverges

Applying this, the limit of the term of the series as n approaches infinity is:
lim_(n->oo) a_n
=lim_(n->oo) n^2/(n^2+1)
= lim_(n->oo) n^2/(n^2(1+1/n^2))
=lim_(n->oo)1/(1+1/n^2)
=1/1+0
=1
The limit of the series is not zero. Therefore, by the nth-Term Test for Divergence, the series diverges.