All Questions
Tagged with closed-form limits
71
questions
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,...
- 25.7k
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 ...
- 25.7k
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 ...
- 25.7k
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 ...
- 25.7k
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(\...
- 25.7k
0
votes
1
answer
54
views
Find $a$ such that the limit is zero
Problem :
Let $x>0$ then define :
$$f(x)=\left(\left(\frac{1}{x}\right)!\left(x!\right)\right)^{\frac{1}{x+\frac{1}{x}}}$$
Then find $a$ such that :
$$\lim_{x\to\infty}f(x)-\frac{1}{2}\left(\frac{1}...
16
votes
1
answer
453
views
Closed form for $\lim\limits_{n\to\infty}\prod\limits_{k=1}^n{\left(2-\frac{2n^2-\pi^2+8}{n^2}\cos{\frac{(2k-1)\pi}{n}}\right)}$?
I am looking for a closed form for:
$$\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)}$$
(Wolfram suggests that it's approximately 6.17966.)
Context:
I ...
- 25.7k
1
vote
1
answer
61
views
A Regularized Beta function limit: $\lim_{a\to0}\frac{1-\text I_\frac zb(a,b)}a$
The goal is to “generalize” the Exponential Integral $\text{Ei}(x)$ using the Regularized Beta function $\text I_z(a,b)$:
$$f(b,z)=\lim_{a\to0}\frac{1-\text I_\frac zb(a,b)}a$$
Some clues include:
$$\...
- 12.5k
34
votes
0
answers
596
views
An iterative logarithmic transformation of a power series
Consider the following iterative process. We start with the function having all $1$'s in its Taylor series expansion:
$$f_0(x)=\frac1{1-x}=1+x+x^2+x^3+x^4+O\left(x^5\right).\tag1$$
Then, at each step ...
- 47.3k
2
votes
1
answer
91
views
Closed form of the series: $\sum _{n=0}^{\infty }\left(\frac{1}{2^n\left(1+\sqrt[2^n]{x}\right)}\right)$
Came across this in a calc textbook from the 1800s and I can't figure out a way to solve it. Trying to write it in product form by taking the integral didn't work. I also tried adding consecutive ...
- 113
48
votes
1
answer
1k
views
How to evaluate double limit of multifactorial $\lim\limits_{k\to\infty}\lim\limits_{n\to 0} \sqrt[n]{n\underbrace{!!!!\cdots!}_{k\,\text{times}}}$
Define the multifactorial function $$n!^{(k)}=n(n-k)(n-2k)\cdots$$ where the product extends to the least positive integer of $n$ modulo $k$. In this answer, I derived one of several analytic ...
- 27.1k
7
votes
1
answer
342
views
An infinite product for $\frac{\pi}{2}$
Please help prove
$$
\begin{align}
\frac{\pi}{2}&=\left(\frac{1}{2}\right)^{2/1}\left(\frac{2^{2}}{1^{1}}\right)^{4/(1\cdot 3)}\left(\frac{1}{4}\right)^{2/3}\left(\frac{2^{2}\cdot4^{4}}{1^{1}\...
- 3,557
5
votes
1
answer
254
views
Evaluate $\lim\limits_{n\to\infty} \frac{\sin(1)+\sin^2(\frac{1}{2})+\ldots+\sin^n(\frac{1}{n})}{\frac{1}{1!}+\frac{1}{2!}+\ldots+\frac{1}{n!}}$
This was a recent problem on the Awesome Math Problem Column. The solution is given as follows:
We shall use Stolz-Cesaro Lemma. We have:
$$\lim_{n\to\infty} \frac{\sin(1)+\sin^2(\frac{1}{2})+\ldots+\...
- 519
8
votes
1
answer
326
views
Challenging limit: $\lim_{\alpha\to0^{+}}\left(\frac{1}{2\alpha}-\int_1^\infty\frac{dx}{\sinh(\pi\alpha x)\sqrt{x^2-1}}\right)$
Here is a challenging limit proposed by a friend:
$$\lim_{\alpha\to0^{+}}\left(\frac{1}{2\alpha}-\int_1^\infty\frac{dx}{\sinh(\pi\alpha x)\sqrt{x^2-1}}\right)$$
and he claims that the closed form ...
- 25.8k
1
vote
2
answers
158
views
Finding a closed form to a minimum of a function
It's a try to find a closed form to the minimum of the function :
Let $0<x<1$ then define :
$$g(x)=x^{2(1-x)}+(1-x)^{2x}$$
Denotes $x_0$ the abscissa of the minimum .
Miraculously using Slater's ...