The Test for Divergence for infinite series (also called the “$\displaystyle {n}$-th term test for divergence of a series”) says that:

$\displaystyle \lim_{{{n}\to\infty}}{a}_{{n}}\ne{0}\ \ \Rightarrow\ \ {\sum_{{{n}={1}}}^{{\infty}}}{a}_{{n}}$ diverges

Notice that this test tells us nothing about $\displaystyle {\sum_{{{n}={1}}}^{{\infty}}}{a}_{{n}}$ if $\displaystyle \lim_{{{n}\to\infty}}{a}_{{n}}={0}$; in that situation the series might converge or it might diverge.

Consider the series $\displaystyle {\sum_{{{n}={1}}}^{\infty}}\frac{{{3}{n}^{{4}}}}{{{4}{n}^{{5}}+{4}}}$

The Test for Divergence tells us that this series: