All Questions
Tagged with closed-form limits
71
questions
4
votes
0
answers
83
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:
$$ \lim_{k \rightarrow \infty} \left[\sum_{a^2+b^2 \le k^2; (a,b) \ne 0} \frac{1}{a^2+b^2} - 2\pi\ln(k) \...
- 17.4k
1
vote
1
answer
145
views
Evaluate $\lim_{n\to\infty}\prod_{k=1}^n \frac{2n}{e}(\arcsin(\frac{k}{n})-\arcsin(\frac{k-1}{n}))$
I'm trying to evaluate $L=\lim\limits_{n\to\infty}f(n)$ where
$$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)$$
We have:
$f(1)\...
- 25.6k
2
votes
0
answers
140
views
closed form for limit?
Consider the function
$$ 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} $$
I want to find ...
- 946
2
votes
1
answer
164
views
Another weird limit involving gamma and digamma function via continued fraction
Context :
I want to find a closed form to :
$$\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$$
Some ...
2
votes
1
answer
99
views
Find the limit and integral $\lim_{\epsilon \to 0} \int_{\epsilon}^{1} \frac{x \sqrt{x} \log(x)}{x^4 + x^2 + 1} \, dx $
Find the limit and integral$$ \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?
...
- 25.6k
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 ...
- 1,686
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)$$
Where $id(x)$ is the identity function mapping all values to itself.
Let $*^{[y]}$ ...
- 16.3k
1
vote
0
answers
72
views
Fibonacci like sequence $f(n) = f(n-1) + f(n-2) + f(n/2)$ and closed form limits?
Consider
$$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)$$
where we take the integer part of the ...
- 16.3k
0
votes
1
answer
62
views
The limit of a Nasty Summation
I'm trying to evaluate the limit as h approachs 0 of the sum from k = 0 to n of:
$\frac{1}{h^n}(-1)^{k+n}\binom{n}{k}\frac{1}{(x+kh)^2-2(x+kh)+17}$
If it helps, it's the limit definition of the nth ...
- 79
1
vote
1
answer
133
views
Does $\lim_{x\to 0} \left(2^{1-x!}3^{1-x!!}4^{1-x!!!}5^{1-x!!!!}6^{1-x!!!!!}\cdot\cdot\cdot\right)^{\frac{1}{x}}=L$ admits a closed form?
I try to simplify this limit :
$$\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 ...
- 12.4k
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 ...
- 25.6k
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 $...
- 25.6k