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 Post subject: On an evaluation of an arctan limit Posted: Sun Oct 29, 2017 7:37 pm
 Administrator  Joined: Sat Nov 07, 2015 6:12 pm
Posts: 841
Location: Larisa
Evaluate the limit

$$\Omega = \lim_{n \rightarrow +\infty} \sum_{k=1}^{n} \frac{\frac{1}{n} \arctan \left ( \frac{k}{n} \right )}{1+2\sqrt{1+\frac{1}{n} \arctan \left ( \frac{k}{n} \right )}}$$

Dan Sitaru

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Imagination is much more important than knowledge. Top   Post subject: Re: On an evaluation of an arctan limit Posted: Sat May 26, 2018 1:02 pm

Joined: Sat Nov 14, 2015 6:32 am
Posts: 157
Location: Melbourne, Australia
Why not prove the more general result?

Let $f:[0, 1] \rightarrow (0, +\infty)$ be a bounded integrable function. Then:

$\lim_{n \rightarrow +\infty} \frac{1}{n} \sum_{k=1}^{n} \frac{f\left ( \frac{k}{n} \right )}{1+2\sqrt{\frac{1}{n} f\left ( \frac{k}{n} \right )+1}} = \frac{1}{3} \int_{0}^{1} f(x) \, {\rm d}x$

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$\displaystyle \sum_{n=1}^{\infty}\frac{1}{n^s}= \prod_{p \; \text{prime}}\frac{1}{1-p^{-s}}$

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