Question

How you solve this?: `lim_(n->oo)sum_(k=1)^n(2k+1)/(k^2(k+1)^2)`

Answer 1

`1`

Explanation:

We know that

`(2 k + 1)/(k^2 (k + 1)^2)=1/k^2-1/(k+1)^2` then

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

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