All Questions
Tagged with closed-form limits
71
questions
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
1
vote
0
answers
205
views
Question on series being expressed in closed form
Given an integer $k$ and $0\leq \alpha \leq 1$, let $f_1(\alpha)=1/k$ and $f_{i+1}(\alpha)=\frac{(k-1)f_i(\alpha) + (f_i(\alpha)^{1/\alpha} + 1)^\alpha}{k}$.
Consider the function $g(\alpha) = \lim_{...
- 11
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
5
votes
2
answers
283
views
Closed formula for the asymptotic limit of a definite integral
I would like to solve the following integral:
$$ I_0 (a,b)= \int_0^1 dx\int_0^{1-x} dz \frac{1}{a z (z-1)+a x z + x(1-b)}$$
in the limit where $b$ is small ($a$ and $b$ are positive constants).
...
- 87
8
votes
1
answer
9k
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A long nasty limit problem: $\lim_{x \to \infty}\left[8e\,\sqrt[\Large x]{x^{x+1}(x-1)!}- 8x^2-4x \ln x - \ln^2 x - (4x + 2 \ln x) \ln 2\pi\right]$
Does the following limit admit a closed-form?
$$\lim_{x \to \infty}\left[8e\,\sqrt[\Large x]{x^{x+1}(x-1)!}- 8x^2-4x \ln x - \ln^2 x - (4x + 2 \ln x) \ln 2\pi\right]$$
My professor gives this ...
- 11k
4
votes
2
answers
215
views
Evaluating $\lim_{x \to 0}\frac{(1+x)^{1/x} - e}{x}$ [duplicate]
How to evaluate the following limit? $$\lim_{x \to 0}\frac{(1+x)^{1/x} - e}{x}.$$
- 1,301
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
3
votes
1
answer
198
views
Limit at Infinity $\lim\limits_{m\to\infty}\frac{\sum\limits_{k=1}^m\cot^{2n+1}\left(\frac{k\pi}{2m+1}\right)}{m^{2n+1}}$
How can I prove the following equality?
$$\lim_{m\to{\infty}}\frac{\displaystyle\sum_{k=1}^m\cot^{2n+1}\left(\frac{k\pi}{2m+1}\right)}{m^{2n+1}}=\frac{2^{2n+1}\zeta(2n+1)}{\pi^{2n+1}}$$
- 5,626
4
votes
3
answers
835
views
The value of the $\lim_{n\rightarrow \infty}\frac{1}{n}\sum_{k=1}^{n}\frac{\sqrt[k]{k!}}{k}$
What is the value of $$\lim_{n\rightarrow \infty}\frac{1}{n}\sum_{k=1}^{n}\frac{\sqrt[k]{k!}}{k}?$$
- 3,167
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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