MHB Does the Series $\sum_{n=2}^{\infty} (-1)^n \frac{4}{5\ln{n}}$ Converge?

[karush]

$\tiny{10.6.10}\\ $
$\textsf{ converge or diverge?}\\$
\begin{align*}\displaystyle
S_n&= \sum_{n=2}^{\infty} (-1)^n \frac{4}{5\ln{n}}\\
&\frac{4}{5} \sum_{n=2}^{\infty} (-1)^n
\frac{1}{\ln{n}}=
\end{align*}
?
??
?

$\textit{converges: alternating series test}$

 

[Prove It]

$\tiny{10.6.10}\\ $
$\textsf{ converge or diverge?}\\$
\begin{align*}\displaystyle
S_n&= \sum_{n=2}^{\infty} (-1)^n \frac{4}{5\ln{n}}\\
&\frac{4}{5} \sum_{n=2}^{\infty} (-1)^n
\frac{1}{\ln{n}}=
\end{align*}
?
??
?

$\textit{converges: alternating series test}$

Yes it converges as ln(n) is a decreasing function.