Answer
See below.
Work Step by Step
If we take Theorem 7,
"If $\displaystyle \sum_{n=1}^{\infty}a_{n}$ converges, then $\displaystyle \lim_{n\rightarrow\infty}a_{n}=0$,"
then, the contraposition of the theorem (equivalent to the theorem),
"If $\displaystyle \lim_{n\rightarrow\infty}a_{n}$ either does not exist or is not 0, then $\displaystyle \sum_{n=1}^{\infty}a_{n}$ diverges,"
is the $n^{th}$ term test for divergence.
The idea behind the test is to quickly see whether we can eliminate the possibility of the series being convergent.
If the test fails, that is, if $\displaystyle \lim_{n\rightarrow\infty}a_{n}=0$, we can not conclude anything. Other tools must be used to further investigate the convergence/divergence of the series.
