lim-n-r-1-4n-n-r-3-r-4-n-2-

Question Number 101373 by Dwaipayan Shikari last updated on 02/Jul/20

\lim_{n \to \infty} \underset{r = 1}{\sum^{4 n}} \frac{\sqrt{n}}{\sqrt{r} {(3 \sqrt{r} + 4 \sqrt{n})}^{2}}

Commented by Dwaipayan Shikari last updated on 02/Jul/20

\frac{1}{n} \lim_{n \to \infty} \underset{r = 1}{\sum^{4 n}} {(\frac{n}{r})}^{\frac{3}{2}} . \frac{1}{{(3 + 4 \sqrt{\frac{n}{r}})}^{2}} = \int_{0}^{4} \frac{1}{x^{\frac{3}{2}}} . \frac{1}{{(3 + \frac{4}{x^{\frac{1}{2}}})}^{2}} d x = \int_{0}^{4} \frac{\frac{1}{x^{\frac{3}{2}}}}{{(3 + \frac{4}{x^{\frac{1}{2}}})}^{2}} d x

- \frac{1}{2} \int_{0}^{4} \frac{\frac{- 2}{x^{\frac{3}{2}}}}{{(3 + \frac{4}{x^{\frac{1}{2}}})}^{2}} d x = - \frac{1}{2} \int_{\infty}^{5} \frac{d t}{t^{2}} = \frac{1}{2} {[\frac{1}{t}]}_{\infty}^{5} = \frac{1}{10}

I s i t r i g h t ? ? ? ? ? ? ?