ratio test and divergence 
I need to be able to prove that this series converges. I know I need to use the ratio test but I do not know how to go about doing it.
Any help is much appreciated! thank you
 A: It’s possible to use the ratio test, though it probably isn’t the easiest approach:
$$\begin{align*}
\frac{\left(\frac{k+1}{k+2}\right)^{(k+1)^2}}{\left(\frac{k}{k+1}\right)^{k^2}}&=\frac{(k+1)^{2k^2}}{k^{k^2}(k+2)^{k^2}}\cdot\left(\frac{k+1}{k+2}\right)^{2k+1}\\\\
&=\left(\frac{k^2+2k+1}{k^2+2k}\right)^{k^2}\cdot\left(\frac{k+1}{k+2}\right)^{2k+1}\\\\
&=\left(1+\frac1{k^2+2k}\right)^{k^2}\cdot\left(1-\frac1{k+2}\right)^{2k+1}\\\\
&=\left(\left(1+\frac1{k^2+2k}\right)^{k^2+2k}\right)^{\frac{k}{k+2}}\cdot\left(\left(1-\frac1{k+2}\right)^{k+2}\right)^{\frac{2k+1}{k+2}}\;.
\end{align*}$$
For large $k$ the quantities
$$\left(1+\frac1{k^2+2k}\right)^{k^2+2k}\quad\text{ and }\quad\left(1-\frac1{k+2}\right)^{k+2}$$
have approximately what numerical values?
A: In fact
\begin{eqnarray*}
L&=&\lim_{k\to\infty}\frac{a_{k+1}}{a_k}=\lim_{k\to\infty}\left(\frac{k+1}{k+2}\right)^{(k+1)^2}\left(\frac{k+1}{k}\right)^{k^2}\\
&=&\lim_{k\to\infty}\left(\frac{k+1}{k+2}\right)^{k^2}\left(\frac{k+1}{k}\right)^{k^2}\left(\frac{k+1}{k+2}\right)^{2k+1}\\
&=&\lim_{k\to\infty}\left(\frac{(k+1)^2}{k(k+2)}\right)^{k^2}\left(1-\frac{1}{k+2}\right)^{2k+1}\\
&=&\lim_{k\to\infty}\left(1+\frac{1}{k(k+2)}\right)^{k^2}\left(1-\frac{1}{k+2}\right)^{2k+1}\\
&=&\lim_{k\to\infty}\left(1+\frac{1}{k(k+2)}\right)^{k(k+2)\frac{k^2}{k(k+2)}}\left(1-\frac{1}{k+2}\right)^{(k+2)\frac{2k+1}{k+2}}\\
&=&e\times\frac{1}{e^2}=\frac{1}{e}<1.
\end{eqnarray*}
So the series converges. You also can use the root test to get $L=\frac{1}{e}$ which is much easier to calculate.
