Question

Calculate `lim_(n->oo)sum_(k=1)^n n/(n^2+k^2) =` ?

Answer 1

`pi/4`

Explanation:

`sum_(k=1)^n n/(n^2+k^2) = sum_(k=1)^n(1/(1+(k/n)^2))1/n`

this is the Riemann approximation for the integral

`lim_(n->oo)sum_(k=1)^n(1/(1+(k/n)^2))1/n = int_0^1 (d xi)/(1+ xi^2) = pi/4`