Show whether the sequence converges (sequence given as a sum)

Question

Here is the problem:
Show whether $a_n$ converges:
$$ frac1{2^2}+frac2{3^2}+dots+frac n{(n+1)^2}$$
So what I tried to do is I tried to show that $a_{n+1}-a_n>0$ which can be simply proven given that $n>0$. What I am having trouble is proving if $a_n$ has an upper boundary. I tried fractional it to partial fractions and the best I can get is the sum of two separate sequences which I don't think does anything useful.
Can anyone help?

Jane · Answer

Using that $kgeq frac{k+1}{2}$ for all positive integers, one gets
$$
a_n = sum_{k=1}^{n} frac{k}{(k+1)^2} geq frac{1}{2}sum_{k=1}^{n} frac{1}{k+1}.
$$
The lower bound sequence obviously diverges, and so the initial sequence diverges as well!

Bernard · Answer

This is immediate with (elementary) asymptotic analysis:
$$frac n{(n+1)^2}sim_inftyfrac n{n^2}=frac1n,$$
and the harmonic series diverges.

Show whether the sequence converges (sequence given as a sum)

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