How you solve this? $lim_(n->oo)sum_(k=1)^n1/((k+1)sqrt(k)+ksqrt(k+1)$

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Answer 1 · 2017-03-18T22:37:22

$1$

$1/((k+1)sqrt(k)+ksqrt(k+1))=((k+1)sqrt(k)-ksqrt(k+1))/((k+1)^2k-k^2(k+1))$

or

$1/((k+1)sqrt(k)+ksqrt(k+1))=(sqrt(k)(k+1)-ksqrt(k+1))/(k(k+1))$

so

$sum_(k=1)^n1/((k+1)sqrt(k)+ksqrt(k+1))=sum_(k=1)^n 1/sqrt(k)-sum_(k=1)^n 1/sqrt(k+1) = 1-1/sqrt(n+1)$

then

$lim_(n->oo)sum_(k=1)^n1/((k+1)sqrt(k)+ksqrt(k+1))=1-lim_(n->oo)1/sqrt(n+1) = 1$

How you solve this? lim_(n->oo)sum_(k=1)^n1/((k+1)sqrt(k)+ksqrt(k+1)lim_(n->oo)sum_(k=1)^n1/((k+1)sqrt(k)+ksqrt(k+1)