$lim_(n->oo)((sqrt(1)+sqrt(2)+sqrt(3)+cdots+sqrt(n))/(n sqrt(n)))$ ?

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Answer 1 · 2016-09-06T09:15:01

$2/3$

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From the above figure we can see that

$sum_(k=1)^n sqrt(k) > int_(x=0)^(x=n) sqrt(x)dx = 2/3n^(3/2)$

enter image source here
and from the above figure we see that

$sum_(k=1)^(n-1)sqrt(k) < int_(x=0)^(x=n) sqrt(x) dx=2/3n^(3/2)$

then follows

$2/3 n^(3/2)-1/sqrt(n) < 2/3n^(3/2) < sum_(k=1)^nsqrt(k) < 2/3n^(3/2)+1/sqrt(n)$

Finally

$2/3-1/n < 1/(nsqrtn)sum_(k=1)^nsqrt(k) < 2/3 + 1/n$

and as $n->oo$ we have

$1/(nsqrtn)sum_(k=1)^nsqrt(k) =2/3$

lim_(n->oo)((sqrt(1)+sqrt(2)+sqrt(3)+cdots+sqrt(n))/(n sqrt(n)))lim_(n->oo)((sqrt(1)+sqrt(2)+sqrt(3)+cdots+sqrt(n))/(n sqrt(n))) ?