# convergence – Prove the sequence \$ sum_ {k = 0} ^ n frac {1} {(n + k) ^ 2} \$ is convergent

I have trouble proving that the sequence $${a_n }$$ whose general term $$a_n$$ is

$$sum_ {k = 0} ^ n frac {1} {(n + k) ^ 2}$$

is convergent. I try to prove it by definition, that is by finding a lower / upper bound and proving that it decreases / increases using induction.

By subtracting $$a_n$$ of $$a_ {n + 1}$$ we get the following for $$n = m$$ (if I'm not mistaken):

$$frac {1} {(2m + 1) ^ 2} + frac {1} {(2m + 2) ^ 2} – frac {1} {m ^ 2}$$

Who is less than $$0$$ for $$n = 1$$ from which I assumed that the sequence is down and therefore trying to prove that $$a_ {n + 1} – a_n <0$$ using induction. But I'm terribly stuck! Thank you in advance for any suggestion.