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Evaluate $\lim_{n\to\infty}\prod_{k=1}^n \frac{2n}{e}(\arcsin(\frac{k}{n})-\arcsin(\frac{k-1}{n}))$
I'm trying to evaluate $L=\lim\limits_{n\to\infty}f(n)$ where
$$f(n)=\prod\limits_{k=1}^n \frac{2n}{e}\left(\arcsin\left(\frac{k}{n}\right)-\arcsin\left(\frac{k-1}{n}\right)\right)$$
We have:
$f(1)\...
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Special property of circle with radius 0.975399...
$4n$ points are uniformly distributed on a circle. Parabolas are drawn in the manner shown below with example $n=4$.
The parabolas' vertices are at the center of the circle. The parabolas have a ...
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