Does sequence ${ s }_{ n }=\sum _{ k=0 }^{ n }{ \frac { 1 }{ { 1+k }^{ 2 } } } $ converge I'm reviewing some Calculus 1 convergence stuff. I want to decide wether ${ s }_{ n }:=\sum _{ k=0 }^{ n }{ \frac { 1 }{ { 1+k }^{ 2 } }  } $
is convergent or not.
Since the sequence is monotnically increasing and has an upper and lower bound it is convergent, but im failing to make the right estimates

1) Sequence is monotonically increasing .
2)Need to find an upper bound:
$\sum_{k=1}^{n} \dfrac{1}{1+k^2}\le$
$\sum_{k=1}^{n}\dfrac{1}{k^2}$.
Consider:
$\sum_{k=2}^{n}\dfrac{1}{k^2} \le$
$\sum_{k=1}^{n-1}\dfrac{1}{k(k+1)} =$
$\sum_{k=1}^{n-1}[\dfrac{1}{k} - \dfrac{1}{k+1}]=$
$1 - \dfrac{1}{n}.$
Bounded above.
Sum converges.
