$N^{th}$ Term Divergence Test

For each problem, use the nth term divergence test to see if the series diverges before any further analysis. If the series diverges say so$.$ If the series does not clearly diverge, then just say that the result is not clear $($there is nothing that this test does to find convergence$).$

$--$All you are doing is testing if the limit of the terms for each series goes to $0$ as $n$ goes to infinity. If the terms don't go to $0,$ then the series diverges. That's it $-$ no other analysis, no other conclusions.

${\displaystyle \underset{n=1}{\overset{}{}}}\frac{2}{n^{\frac{2}{3}}}$

${\displaystyle \underset{n=1}{\overset{}{}}}0\mathrm{.}5\frac{1}{n^2}$

${\displaystyle \underset{n=0}{\overset{}{}}}sin(\frac{n}{2})$

${\displaystyle \underset{n=2}{\overset{}{}}}\frac{2}{94n}$

${\displaystyle \underset{n=1}{\overset{}{}}}\frac{10}{n^2}\frac{20}{n^3}$

${\displaystyle \underset{n=-10}{\overset{}{}}}5e^{\frac{n}{20}}$

${\displaystyle \underset{n=1}{\overset{}{}}}n(0\mathrm{.}1{)}^n$

${\displaystyle \underset{n=-2}{\overset{}{}}}\frac{5n^2-7n}{-3n^{2}1000}$

${\displaystyle \underset{n=4}{\overset{}{}}}{n}^2{e}^{-n}$