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4
questions
5
votes
0
answers
131
views
Infinite product of areas in a square, inscribed quarter-circle and line segments.
The diagram shows a square of area $An$ and an enclosed quarter-circle.
Line segments are drawn from the bottom-left vertex to points that are equally spaced along the quarter-circle.
The regions ...
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3
votes
0
answers
121
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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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6
votes
2
answers
270
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What is a closed form of this limit? (product of areas in circle with parabolas)
I am looking for a closed form of
$L=\lim\limits_{n\to\infty}\prod\limits_{k=1}^n \left(\left(6+\frac{4n}{\pi}\left(\sin{\frac{\pi}{2n}}\right)\left(\cos{\frac{(2k-1)\pi}{2n}}\right)\right)^2-\left(\...
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16
votes
1
answer
453
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Closed form for $\lim\limits_{n\to\infty}\prod\limits_{k=1}^n{\left(2-\frac{2n^2-\pi^2+8}{n^2}\cos{\frac{(2k-1)\pi}{n}}\right)}$?
I am looking for a closed form for:
$$\lim_{n\to\infty}\prod_{k=1}^n{\left(2-\frac{2n^2-\pi^2+8}{n^2}\cos{\frac{(2k-1)\pi}{n}}\right)}$$
(Wolfram suggests that it's approximately 6.17966.)
Context:
I ...
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