sum_(n=1)^oo n/(2n+3) Verify that the infinite series diverges

sum_(n=1)^oo n/(2n+3)
To verify if the series diverges, apply the nth-Term Test for Divergence. 
It states that if the sequence a_n does not converge to zero, then the series diverges.

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

 
Applying this, the limit of the term of the series as n approaches infinity is:
lim_(n->oo) n/(2n + 3)
=lim_(n->oo) n/(n(2+3/n))
= lim_(n->oo) 1/(2+3/n)
=(lim_(n->oo)1)/(lim_(n->oo) (2+3/n))
= 1/(2+0)
=1/2
The limit of the series is not zero. Therefore, by the  nth-Term Test for Divergence, the series diverges.