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

sum_(n=1)^oo n/(n+1)
To verify if this infinite series diverges, apply the Divergent Test.
In the Divergence Test, 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

So, taking the limit of a_n as n approaches infinity yields:
lim_(n->oo) a_n
=lim_(n->oo) n/(n+1)
=lim_(n->oo) n/(n(1+1/n))
=lim_(n->oo) 1/(1+1/n)
=1/(1+0)
=1
Since the result is not equal to zero, therefore, the series is divergent.