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$\lim\limits_{n\to\infty}\dfrac{\sqrt1+\sqrt2+\sqrt3+\ldots+\sqrt n}{n\sqrt n}$

I am having trouble doing this problem. I have attempted to take the Riemann Sum but cannot get past the square root. I also tried to upper-bound and lower-bound it, but I got stuck doing this.

Angelo
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Ritvik Bordia
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  • Welcome to MSE. Please use MathJax to format your posts. To begin with, surround math expressions (including numbers) with $ signs and use _ for subscripts. $x_1$ comes out as $x_1$. – saulspatz Mar 05 '21 at 03:19
  • See also https://math.stackexchange.com/questions/1172144/problem-on-limits-lim-n-to-infty-frac-sqrt1-sqrt2-sqrt3-sqrt4, https://math.stackexchange.com/questions/2114065/limit-lim-n-to-infty-n-3-21-sqrt2-ldots-sqrtn-lim-n-to-infty?noredirect=1, https://math.stackexchange.com/questions/2126922/finding-lim-n-rightarrow-infty-frac-sqrt1-sqrt2-sqrt3-cdots-sqr?noredirect=1, https://math.stackexchange.com/questions/2149746/lim-n-to-infty-frac-sqrtn-sqrt1-sqrt2-sqrtnn2?noredirect=1 – player3236 Mar 05 '21 at 03:23

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