All Questions
25
questions
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
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
$$
\lim_{\epsilon \to 0} \int_{\epsilon}^{1} \frac{x \sqrt{x} \log(x)}{...
1
vote
1
answer
61
views
A Regularized Beta function limit: $\lim_{a\to0}\frac{1-\text I_\frac zb(a,b)}a$
The goal is to “generalize” the Exponential Integral $\text{Ei}(x)$ using the Regularized Beta function $\text I_z(a,b)$:
$$f(b,z)=\lim_{a\to0}\frac{1-\text I_\frac zb(a,b)}a$$
Some clues include:
$$\...
- 12.5k
2
votes
1
answer
938
views
A limit of combination
I want to find the closed form of the limit,
\begin{align*}
I(k,r):=\lim_{x\rightarrow 0}\left\{\sum\limits_{j=1}^{r+2-k} (-1)^{r+3-j-k} \binom{r-j}{k-2}\frac{1}{x^j}+\frac{1}{(1+x)^{k-1}x^{r-k+2}}\...
- 537
1
vote
0
answers
79
views
Find the limit: $\lim_{n\to\infty} \left (1+ \frac {1}{2^2}+ \frac {1}{3^3}+...+\frac {1}{n^n} \right)$ [duplicate]
Does the limit:
$$\lim_{n\to\infty} \left (1+ \frac {1}{2^2}+ \frac {1}{3^3}+...+\frac {1}{n^n} \right)$$
admit a closed form?
I know only Riemann-Zeta function.I've just discovered this sum.
- 477
2
votes
4
answers
155
views
Find closed formula and limit for $a_1 =1$, $2a_{n+1}a_n = 4a_n + 3a_{n+1}$
Tui a sequence $(a_n)$ defined for all natural numbers given by
$$a_1 =1, 2a_{n+1}a_n = 4a_n + 3a_{n+1}, \forall n \geq 1$$
Find the closed formula for the sequence and hence find the limit.
Here, ...
- 157
39
votes
3
answers
2k
views
What's the limit of $\sqrt{2 + \sqrt{2-\sqrt{2+\sqrt{2-\sqrt{2-\sqrt{2 + ...}}}}}} $?
Let's look at the continued radical
$ R = \sqrt{2 + \sqrt{2-\sqrt{2+\sqrt{2-\sqrt{2-\sqrt{2 + ...}}}}}} $
whose signs are defined as $ (+, -, +, -, -, + ,-, -, -,...)$, similar to the sequence $...
user346361
4
votes
2
answers
128
views
Let $\sum_{n=0}^\infty \frac{(-1)^{n+1}}{3 n+6 (-1)^n}$, does it converge or does it diverge?
Let $\displaystyle \sum_{n=0}^\infty \frac{(-1)^{n+1}}{3 n+6 (-1)^n}$, does it converge or does it diverge?
I'm not completely sure that my calculation is correct, check it please.
$$\begin{align}\...
- 30.8k
2
votes
1
answer
57
views
Closed form and limit of the sequence $a_{n+1}=\frac{-5a_n}{2n+1}$
I have no idea about how to deal with point B. Can anyone help me? Also, an elegant way to solve point A would be great but it's not that important. Thanks in advance for the help!
A) Suppose $a\in\...
- 31
1
vote
4
answers
95
views
How to solve $\lim_{x \to 0} \dfrac{\arctan(x)}{e^{2x}-1}$?
$$\lim_{x \to 0} \dfrac{\arctan(x)}{e^{2x}-1}$$
I have no idea how to do this, initially I thought that perhaps $\lim_{x \to 0} \dfrac{e^x-1}{x} = 1$ might be of use but I don't see how I can rewrite ...
3
votes
2
answers
60
views
Evaluation of $\lim_{n \to \infty} ((n+1)!\ln (a_n))$
Consider the sequence $(a_n)_{n \geq1}$ such that $a_0=2$ and $a_{n-1}-a_n=\frac{n}{(n+1)!}$. Evaluate $$\lim_{n \to \infty} ((n+1)!\ln (a_n))$$
Could someone hint me as how to achieve value of $a_n$ ...
- 1,367
1
vote
1
answer
59
views
Finding $\lim_{n\to \infty} \frac{1}{n} \sum_{k=1}^n \frac{a^{1+\frac{k}{n}}}{a^{1+\frac{k}{n}}+1} $
As the question says,
$$\lim_{n\to \infty} \frac{1}{n} \sum_{k=1}^n \frac{a^{1+\frac{k}{n}}}{a^{1+\frac{k}{n}}+1} $$
where a is a constant, $a>0$.
3
votes
1
answer
204
views
A limit using the Euler number: $\lim_{n\rightarrow\infty}\frac{n!}{(n-i)!}\left(\frac{c}{n}\right)^{n-i}$
What is answer of this limit and how can I get it? $c$ and $i$ are constants.
$$\lim_{n\rightarrow\infty}\frac{n!}{(n-i)!}\left(\frac{c}{n}\right)^{n-i}$$
I guess it will envolve some Neper/the Euler ...
- 579
1
vote
1
answer
59
views
Finding a limit of a two variable function: $f(x,y)=\frac {\sin(x^2-xy)}{\vert x\vert} $
I have this exercise but not sure if I'm doing it right
$$\lim_{(x,y)\to (0,0)} \frac {\sin(x^2-xy)}{\vert x\vert} $$
I assume $\frac {\sin(x^2-xy)}{\vert x\vert}\le\frac {1}{\vert x \vert} $
then ...
- 11
6
votes
1
answer
261
views
Find $\lim_{a\to \infty}\frac{1}{a}\int_0^{\infty}\frac{x^2+ax+1}{1+x^4}\cdot\arctan(\frac{1}{x})dx$
Find
$$
\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