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## On an evaluation of an arctan limit

Real Analysis
Tolaso J Kos
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### On an evaluation of an arctan limit

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
Imagination is much more important than knowledge.
Riemann
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### Re: On an evaluation of an arctan limit

Why not prove the more general result?

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

$\lim_{n \rightarrow +\infty} \frac{1}{n} \sum_{k=1}^{n} \frac{f\left ( \frac{k}{n} \right )}{1+2\sqrt{\frac{1}{n} f\left ( \frac{k}{n} \right )+1}} = \frac{1}{3} \int_{0}^{1} f(x) \, {\rm d}x$
$\displaystyle \sum_{n=1}^{\infty}\frac{1}{n^s}= \prod_{p \; \text{prime}}\frac{1}{1-p^{-s}}$