Wednesday, 28 December 2016

calculus - Evaluate limlimitsntoinftysumlimitsnk=0dfracsqrtnn+k2(n=1,2,cdots)

I tried to change it into a Riemann sum but failed, since




lim
which is not a standard form. Maybe, it need apply the squeeze theorem, but how to evaluate the bound.



By the way, WA gives its result
\begin{align*} \lim_{n \to \infty}\sum_{k=0}^n \frac{\sqrt{n}}{n+k^2}=\frac{\pi}{2}. \end{align*}

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