All Questions
Tagged with closed-form limits
71
questions
6
votes
1
answer
143
views
Is there a closed form for the quadratic Euler Mascheroni Constant?
Short Version:
I am interested in computing (as a closed form) the limit if it does exist:
$$ \lim_{k \rightarrow \infty} \left[\sum_{a^2+b^2 \le k^2; (a,b) \ne 0} \frac{1}{a^2+b^2} - 2\pi\ln(k) \...
- 17.5k
1
vote
1
answer
145
views
Evaluate $\lim_{n\to\infty}\prod_{k=1}^n \frac{2n}{e}(\arcsin(\frac{k}{n})-\arcsin(\frac{k-1}{n}))$
I'm trying to evaluate $L=\lim\limits_{n\to\infty}f(n)$ where
$$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)$$
We have:
$f(1)\...
- 25.7k
2
votes
0
answers
141
views
closed form for limit?
Consider the function
$$ f(x)=\lim_{k \to \infty}\bigg(\int_0^x \sum_{n=1}^k e^{\frac{\log n}{\log r}}~dr \bigg)\bigg( \int_0^1 \sum_{n=1}^k e^{\frac{\log n}{\log r}}~dr \bigg)^{-1} $$
I want to find ...
- 756
2
votes
1
answer
164
views
Another weird limit involving gamma and digamma function via continued fraction
Context :
I want to find a closed form to :
$$\lim_{x\to 0}\left(\frac{f(x)}{f(0)}\right)^{\frac{1}{x}}=L,f(x)=\left(\frac{1}{1+x}\right)!×\left(\frac{1}{1+\frac{1}{1+x}}\right)!\cdots$$
Some ...
2
votes
1
answer
100
views
Find the limit and integral $\lim_{\epsilon \to 0} \int_{\epsilon}^{1} \frac{x \sqrt{x} \log(x)}{x^4 + x^2 + 1} \, dx $
Find the limit and integral$$ \lim_{\epsilon \to 0} \int_{\epsilon}^{1} \frac{x \sqrt{x} \log(x)}{x^4 + x^2 + 1} \, dx $$
My try
$$
\lim_{\epsilon \to 0} \int_{\epsilon}^{1} \frac{x \sqrt{x} \log(x)}{...
12
votes
1
answer
209
views
$e$ is hidden in Pascal's (binomial) triangle. What is hidden in the trinomial triangle, in the same way?
In Pascal's triangle, denote $S_n=\prod\limits_{k=0}^n\binom{n}{k}$. It can be shown that
$$\lim_{n\to\infty}\frac{S_{n-1}S_{n+1}}{{S_n}^2}=e$$
What is the analogous result for the trinomial triangle?
...
- 25.7k
3
votes
4
answers
365
views
What is $\lim_{x\to\infty}\frac{\int_{0}^{x}\cos\{t-\cos t\}dt}{x}$?
I want to find a closed form for the average value of $\cos\{t-\cos t\}$ where $\{n\}$ denotes the fractional part of $n$. I do not have experience finding an average value over an infinite domain but ...
- 1,688
1
vote
2
answers
83
views
Evaluating $\lim_{k\to+\infty}\frac12\sin(\sqrt{k+1})-2 \left(\sin(\sqrt{k+1})-\sqrt{k+1}\cos(\sqrt{k+1})\right)+\sum_{n=0}^k\sin(\sqrt n)$
I need help at evaluating this to some closed form formula:
$$\lim_{k\to+\infty}\frac{\sin\left(\sqrt{k+1}\right)}{2}-2 \left(\sin\left(\sqrt{k+1}\right)-\sqrt{k+1}\cos\left(\sqrt{k+1}\right) \right)+\...
- 21
0
votes
0
answers
50
views
Asymptotics for this limit iteration with $f(x)= 2x + x^5 ,g(x) = x + x^3$
Consider $x>0$
Let
$$f(x)= 2x + x^5$$
$$g(x) = x + x^3$$
$$f(r(x))=r(f(x))=id(x)$$
$$g(u(x)) = u(g(x))=id(x)$$
Where $id(x)$ is the identity function mapping all values to itself.
Let $*^{[y]}$ ...
- 16.4k
1
vote
0
answers
72
views
Fibonacci like sequence $f(n) = f(n-1) + f(n-2) + f(n/2)$ and closed form limits?
Consider
$$f(1) = g(1) = 1$$
$$f(2) = A,g(2) = B$$
$$f(3) = 1 + A,g(3) = 1+B$$
And for $n>3$ :
$$f(n) = f(n-1) + f(n-2) + f(n/2)$$
$$g(n) = g(n-1) + g(n-2)$$
where we take the integer part of the ...
- 16.4k
0
votes
1
answer
62
views
The limit of a Nasty Summation
I'm trying to evaluate the limit as h approachs 0 of the sum from k = 0 to n of:
$\frac{1}{h^n}(-1)^{k+n}\binom{n}{k}\frac{1}{(x+kh)^2-2(x+kh)+17}$
If it helps, it's the limit definition of the nth ...
- 87
1
vote
1
answer
133
views
Does $\lim_{x\to 0} \left(2^{1-x!}3^{1-x!!}4^{1-x!!!}5^{1-x!!!!}6^{1-x!!!!!}\cdot\cdot\cdot\right)^{\frac{1}{x}}=L$ admits a closed form?
I try to simplify this limit :
$$\lim_{x\to 0} \left(2^{1-x!}3^{1-x!!}4^{1-x!!!}5^{1-x!!!!}6^{1-x!!!!!}\cdots\right)^{\frac{1}{x}}=L$$
Where we compose the Gamma function with itself .
From the past ...
1
vote
0
answers
100
views
Reduction of $_3\text F_2(a,a,1-b;a+1,a+1;x)$ with the hypergeometric function
A derivative of the incomplete beta function $\text B_x(a,b)$ uses hypergeometric $_3\text F_2$
$$\frac{d\text B_x(a,b)}{da}=\ln(x)\text B_x(a,b)-\frac{x^a}{a^2}\,_3\text F_2(a,a,1-b;a+1,a+1;x)$$
Now ...
- 12.5k
7
votes
1
answer
209
views
What is the product of the areas of every regular polygon inscribed in a circle of area $1$?
What is a closed form of $P=\prod\limits_{k=3}^{\infty}\frac{k}{2\pi}\sin{\left(\frac{2\pi}{k}\right)}\approx 0.05934871...$ ?
This is the product of the areas of every regular polygon inscribed in a ...
- 25.7k
5
votes
1
answer
115
views
Hidden property of the graph of $y=\tan{x}$: infinite product of lengths of zigzag line segments converges, but to what?
On the graph of $y=\tan{x}$, $0<x<\pi/2$, draw $2n$ zigzag line segments that, with the x-axis, form equal-width isosceles triangles whose top vertices lie on the curve. Here is an example with $...
- 25.7k
2
votes
4
answers
270
views
Show that $\lim_{n\to\infty}n\left(n\ln{n}+\ln{\sqrt{2}}-n-\sum_{k=1}^n \ln{\left(k-\frac{1}{2}\right)}\right)=\frac{1}{24}$.
I am trying to show that
$$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 ...