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

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Answer 1 · 2017-05-11T10:49:39

$1$

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

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

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