Answer to Question #350094 in Real Analysis for Nikhil

Question #350094

Evaluate



lim 2r Σ r=1 [2n^2/(n+r)^3]


n→∞

1
Expert's answer
2022-06-13T14:41:03-0400

"\\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}"


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