All Questions
12
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)\...
- 25.7k
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
$$
\lim_{\epsilon \to 0} \int_{\epsilon}^{1} \frac{x \sqrt{x} \log(x)}{...
3
votes
4
answers
365
views
What is $\lim_{x\to\infty}\frac{\int_{0}^{x}\cos\{t-\cos t\}dt}{x}$?
I want to find a closed form for the average value of $\cos\{t-\cos t\}$ where $\{n\}$ denotes the fractional part of $n$. I do not have experience finding an average value over an infinite domain but ...
- 1,688
1
vote
2
answers
83
views
Evaluating $\lim_{k\to+\infty}\frac12\sin(\sqrt{k+1})-2 \left(\sin(\sqrt{k+1})-\sqrt{k+1}\cos(\sqrt{k+1})\right)+\sum_{n=0}^k\sin(\sqrt n)$
I need help at evaluating this to some closed form formula:
$$\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
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
0
votes
0
answers
97
views
Expressing the Golomb-Dickman constant in closed-form
Is there a way to express the Golomb-Dickman constant ($\lambda$) (A084945) in a closed-form expression? Here's the Wikipedia article for the Golomb-Dickman constant, but it's not as useful in my ...
- 1,271
3
votes
2
answers
90
views
Evaluating the limit of the sequence: $\frac{ 1^a + 2^a +..... n^a}{(n+1)^{a-1}[n^2a + n(n+1)/2]}$
My friend gave me this question to solve a few days ago and after I got no way to solve this, I thought I should seek some help.
I had to evaluate the limit of the following when $n$ tends to ...
1
vote
1
answer
59
views
Finding $\lim_{n\to \infty} \frac{1}{n} \sum_{k=1}^n \frac{a^{1+\frac{k}{n}}}{a^{1+\frac{k}{n}}+1} $
As the question says,
$$\lim_{n\to \infty} \frac{1}{n} \sum_{k=1}^n \frac{a^{1+\frac{k}{n}}}{a^{1+\frac{k}{n}}+1} $$
where a is a constant, $a>0$.
6
votes
1
answer
261
views
Find $\lim_{a\to \infty}\frac{1}{a}\int_0^{\infty}\frac{x^2+ax+1}{1+x^4}\cdot\arctan(\frac{1}{x})dx$
Find
$$
\lim_{a\to \infty}
\frac{1}{a}
\int_0^{\infty}\frac{x^2+ax+1}{1+x^4} \arctan\left(\frac{1}{x}\right)dx
$$
I tried to find
$$
\int_0^{\infty} \frac{x^2+ax+1}{1+x^4}\arctan\left(\frac{...
- 4,274
10
votes
3
answers
484
views
Evaluating the limit of a certain definite integral
Let $\displaystyle f(x)= \lim_{\epsilon \to 0} \frac{1}{\sqrt{\epsilon}}\int_0^x ze^{-(\epsilon)^{-1}\tan^2z}dz$ for $x\in[0,\infty)$.
Evaluate $f(x)$ in closed form for all $x\in[0,\infty)$ and ...
- 1
10
votes
3
answers
2k
views
Finding the value of the infinite sum $1 -\frac{1}{4} + \frac{1}{7} - \frac{1}{10} + \frac{1}{13} - \frac{1}{16} + \frac{1}{19} + ... $ [duplicate]
Can anyone help me to find what is the value of $1 -\frac{1}{4} + \frac{1}{7} - \frac{1}{10} + \frac{1}{13} - \frac{1}{16} + \frac{1}{19} + ... $ when it tends to infinity
The first i wanna find the ...
- 379
9
votes
2
answers
653
views
Integral $S_\ell(r) = \int_0^{\pi}\int_{\phi}^{\pi}\frac{(1+ r \cos \psi)^{\ell+1}}{(1+ r \cos \phi)^\ell} \rm d\psi \ \rm d\phi $
Is there a closed form for $|r|<1$ and $\ell>0$ integer?
The solution for the special cases $\ell=2$ and $4$ would also be interesting if the general case is not available.
Integrating ...
- 1,258