Answer to Question #350094 in Real Analysis for Nikhil

Question

Evaluate

lim 2r Σ r=1 [2n^2/(n+r)^3]

n→∞

Answer 1

$\lim\limits_{n\rightarrow\infty}{\sum_{r=1}^{2n}\frac{2n^2}{\left(n+r\right)^3}}=\lim\limits_{n\rightarrow\infty}{\sum_{r=1}^{2n}\frac{2n^2}{{n^3\left(1+\frac{r}{n}\right)}^3}}=\lim\limits_{n\rightarrow\infty}{\frac{1}{n}\sum_{r=1}^{2n}\frac{2}{\left(1+\frac{r}{n}\right)^3}}=$

$=2\int_{\lim\limits_{n\rightarrow\infty}{\frac{1}{n}}}^{2}\frac{dx}{\left(1+x\right)^3}=2\int_{0}^{2}\frac{d\left(1+x\right)}{\left(1+x\right)^3}=$

$=2\left.\frac{\left(1+x\right)^{-2}}{-2}\right|\begin{matrix}2\\0\\\end{matrix}=\left.-\frac{1}{\left(1+x\right)^2}\right|\begin{matrix}2\\0\\\end{matrix}=-\frac{1}{9}+1=\frac{8}{9}$