All Questions
Tagged with closed-form limits
71
questions
4
votes
0
answers
162
views
Determine sum of the series $\sum_{k=1}^\infty\frac{1}{n(n+1)(n+2)}$ [duplicate]
I have the following problem,
$$\sum_{k=1}^\infty\frac{1}{n(n+1)(n+2)}$$
And I try to work as follows:
Hint: Partial Fraction decomposition:
$\begin{aligned}
\frac{1}{n(n+1)(n+2)} &= \frac{...
- 3,919
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
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 ...
4
votes
4
answers
334
views
Calculus - Finding limit (NOT L'Hopital's Rule): $\lim_{x \to 1^-}\frac{x^2+x+\sin({\pi\over 2}x)-3}{x-1}$
How do I find this limit?
$$\displaystyle{\lim_{x \to 1^-}}\frac{x^2+x+\sin({\pi \over 2}x)-3}{x-1}$$
I am unable to factor the numerator to get rid of the denominator. Can someone please help? ...
- 293
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
144
views
$\lim_{z\to -1} (z+1) \sin(\frac{1}{z+1})$, for complex variable $z$.
I want to find this limit for complex variable $z$
$$\lim_{z\to -1} (z+1) \sin(\frac{1}{z+1})$$
In the real case I know $\sin(z)$ is bounded by $-1, 1,$ and the limit is $0$. But in the complex case ...
- 261
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
$$
\int_0^{\infty} \frac{x^2+ax+1}{1+x^4}\arctan\left(\frac{...
- 4,274
1
vote
5
answers
99
views
Limit of a function of two variables: $\lim_{(x,y) \to 0}\dfrac{x^2y}{17x^2+y^2}$
$$\lim_{(x,y) \to 0} \dfrac{x^2y}{17x^2+y^2}$$
I want to obtain this limit but don't know how to. The most general advice I've found is to convert this function into polar coordinates, so when I do ...
- 23
1
vote
3
answers
116
views
Find the limit of the sequence $\left( \sqrt {2n^{2}+n}-\sqrt {2n^{2}+2n}\right) _{n\in N}$
My answer is as follows, but I'm not sure with this:
$\lim _{n\rightarrow \infty }\dfrac {\sqrt {2n^{2}+n}}{\sqrt {2n^{2}+2n}}=\lim _{n\rightarrow \infty }\left( \dfrac {2n^{2}+n}{2n^{2}+2n}\right) ^{\...
- 39
8
votes
4
answers
540
views
A limit related to super-root (tetration inverse).
Recall that tetration ${^n}x$ for $n\in\mathbb N$ is defined recursively: ${^1}x=x,\,{^{n+1}}x=x^{({^n}x)}$.
Its inverse function with respect to $x$ is called super-root and denoted $\sqrt[n]y_s$ (...
- 47.3k
9
votes
1
answer
416
views
Closed Form for $~\lim\limits_{n\to\infty}~\sqrt n\cdot(-e)^{-n}\cdot\sum\limits_{k=0}^n\frac{(-n)^k}{k!}$
$\qquad\qquad\qquad$ Does $~\displaystyle\lim_{n\to\infty}\frac{\sqrt n}{(-e)^n}\cdot\sum_{k=0}^n\frac{(-n)^k}{k!}~$ possess a closed form expression ?
Inspired by this frequently asked question, I ...
- 48.5k
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