I have the following problem given this series:
$ 1\over 5 $ + $ 2^2\over 5^2 $ + $3^2\over 5^3 $ + $4^2 \over 5^4 $
Does it converge or diverge and what test should be used. I have came to the conclusion that the limit approaches + infinity but don't know how to create a proof for that statement and whether it confirms that it diverges. my apologies for the earlier version, that was my poor interpretation of the original problem, which I have updated this post with.
$a_n=\dfrac{n^2}{5^n}$. The sequence converges to $0$, since exponential growth is faster than polynomial.
As for the series, let's try the ratio test: $\mid\dfrac {a_{n+1}}{a_n}\mid=\dfrac 15(\dfrac {n+1}n)^2\to\dfrac 15$. Hence the series converges.