Calculus of a Single Variable, Chapter 9, 9.2, Section 9.2, Problem 12

sum_(n=1)^oo n/sqrt(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) a_n =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/sqrt(n^2+1)
=lim_(n->oo) n/sqrt(n^2(1+1/n^2))
=lim_(n->oo) n/(nsqrt(1+1/n^2))
=lim_(n->oo) 1/sqrt(1+1/n^2)
=(lim_(n->oo)1)/(lim_(n->oo)sqrt(1+1/n^2))
=1/sqrt(0+1)
=1
The limit of the series is not zero. Therefore, by the nth-Term Test forDivergence, the series diverges.