Question

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

Answer 1

`2/3`

Explanation:

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)`

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`