All Questions
13
questions
1
vote
1
answer
145
views
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)\...
- 25.8k
7
votes
1
answer
209
views
What is the product of the areas of every regular polygon inscribed in a circle of area $1$?
What is a closed form of $P=\prod\limits_{k=3}^{\infty}\frac{k}{2\pi}\sin{\left(\frac{2\pi}{k}\right)}\approx 0.05934871...$ ?
This is the product of the areas of every regular polygon inscribed in a ...
- 25.8k
5
votes
1
answer
115
views
Hidden property of the graph of $y=\tan{x}$: infinite product of lengths of zigzag line segments converges, but to what?
On the graph of $y=\tan{x}$, $0<x<\pi/2$, draw $2n$ zigzag line segments that, with the x-axis, form equal-width isosceles triangles whose top vertices lie on the curve. Here is an example with $...
- 25.8k
2
votes
4
answers
270
views
Show that $\lim_{n\to\infty}n\left(n\ln{n}+\ln{\sqrt{2}}-n-\sum_{k=1}^n \ln{\left(k-\frac{1}{2}\right)}\right)=\frac{1}{24}$.
I am trying to show that
$$L=\lim\limits_{n\to\infty}n\left(n\ln{n}+\ln{\sqrt{2}}-n-\sum\limits_{k=1}^n \ln{\left(k-\frac{1}{2}\right)}\right)=\frac{1}{24}$$
Desmos strongly suggests that this is true,...
- 25.8k
5
votes
2
answers
134
views
Find $C$ such that $\frac{1}{n}\prod_{k=1}^{n}C\left(\cos{\frac{k\pi}{2(n+1)}}+\sin{\frac{k\pi}{2(n+1)}}-1\right)$ converges to a positive number.
I'm looking for the value of $C$ such that $L=\lim\limits_{n\to\infty}\frac{1}{n}\prod\limits_{k=1}^{n}C\left(\cos{\frac{k\pi}{2(n+1)}}+\sin{\frac{k\pi}{2(n+1)}}-1\right)$ equals a positive real ...
- 25.8k
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 ...
- 25.8k
3
votes
0
answers
121
views
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 ...
- 25.8k
6
votes
2
answers
270
views
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(\...
- 25.8k
16
votes
1
answer
453
views
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 ...
- 25.8k
7
votes
1
answer
342
views
An infinite product for $\frac{\pi}{2}$
Please help prove
$$
\begin{align}
\frac{\pi}{2}&=\left(\frac{1}{2}\right)^{2/1}\left(\frac{2^{2}}{1^{1}}\right)^{4/(1\cdot 3)}\left(\frac{1}{4}\right)^{2/3}\left(\frac{2^{2}\cdot4^{4}}{1^{1}\...
- 3,557
4
votes
0
answers
72
views
How do I find the finite limits of this infinite product?
What is... $$\lim_{\omega \to \infty}
\left( {1 \over {a^{\omega}}} \cdot \prod_{N=1}^{\omega} (1+e^{b \cdot c^{-N}}) \right)$$
I'd like closed form solutions, and in this case that means any ...
- 8,361
8
votes
2
answers
1k
views
Infinite Product $\prod_{n=1}^\infty\left(1+\frac1{\pi^2n^2}\right)$
How do I find:
$$\prod_{n=1}^\infty\; \left(1+ \frac{1}{\pi ^2n^2}\right) \quad$$
I am pretty sure that the infinite product converges, but if it doesn't please let me know if I have made an error.
...
- 81
24
votes
6
answers
6k
views
Evaluating the infinite product $\prod\limits_{k=2}^\infty \left ( 1-\frac1{k^2}\right)$
Evaluate the infinite product
$$\lim_{ n\rightarrow\infty }\prod_{k=2}^{n}\left ( 1-\frac{1}{k^2} \right ).$$
I can't see anything in this limit , so help me please.
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