How do I test convergence of this series $1+\frac1{2^2}+\frac{2^2}{3^3}+\frac{3^3}{4^4}+\dots$? An infinite positive term series is given :
$$1+\frac{1}{2^2}+\frac{2^2}{3^3}+\frac{3^3}{4^4}+....$$
and I have to test its convergence.
I am having trouble in finding out it's $n^{th}$ term.
I Initially thought about something like $$U_n=\frac{(n-1)^{(n-1)}}{n^n} \text{ for n>1}$$
But my book suggested 
$$U_n=\frac{n^n}{(n+1)\times n +1}$$
Funny thing is, it fails miserably to get the required series.
Can someone suggest a better $n^{th}$ term? And how to proceed further from it. 
Thank You!

You have
$$
U_n=\frac{(n-1)^{n-1}}{n^n}=\frac1n\cdot\left(\frac{n-1}{n}\right)^{n-1}.
$$
A limit familiar from the definition of Napier's constant then leads to a lower bound proving that the series diverges.
