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.