All Questions
224
questions
6
votes
2
answers
257
views
Problematic limit $\epsilon \to 0 $ for combination of hypergeometric ${_2}F_2$ functions
In an earlier question, the integral $$I_n(c)=\int_0^\infty x^n (1+x)^n e^{-n c x^2} dx$$ was considered with particular focus on its behavior for positive integer $n$. In trying to analyze this, it ...
- 16.7k
0
votes
4
answers
196
views
How to evaluate $\sum\limits_{n=3}^ \infty \frac{1}{n \ln(n)(\ln(\ln(n)))^2}$
I saw this problem : Prove that $\sum\limits_{n=3}^ \infty \frac{1}{n \ln(n)(\ln(\ln(n)))^2}$ converges, this is an easy problem could be proved using Cauchy condensation test twice.
$$\sum_{n=3}^ \...
- 6,563
0
votes
1
answer
47
views
Sum sequence using Stolz–Cesàro
I have this sequence, and I need to find the convergence of the sum sequence.
The answer is - sum equal π/4.
But I tried to solve it by Stolz–Cesàro, as you can see in the picture, And what I got is ...
- 215
4
votes
1
answer
125
views
Find value of this sum
Let $$\lim_{x\rightarrow 0}\frac{f^{}(x)}{x}=1$$
and for every $x,y \in \mathbb{R} $ we have:
$$f(x+y)=f(x)-f(y)+ xy(x+y)$$
Now Find :
$$\sum_{i=11}^{17}f^{\prime} (i)$$
I think this question is ...
- 4,697
0
votes
2
answers
200
views
how to calculate $\sum\limits_{k=1}^{+\infty }{\arctan \frac{1}{1+k^{2}}}$
Question: how to calculate $$\sum\limits_{k=1}^{+\infty }{\arctan \frac{1}{1+k^{2}}}$$
My attempt
Let
$\arctan \theta =\frac{i}{2}\ln \left( \frac{i+\theta }{i-\theta } \right)$
$$S=\sum\limits_{k=1}^{...
0
votes
0
answers
104
views
Is this divergent series, convergent?
Examining the series $\sum_{n=1}^{\infty} \frac{1}{nx}$ alongside its integral counterpart reveals insights into its convergence. Notably, the integral over intervals from $10^n$ to $10^{n+1}$ yields ...
- 81
1
vote
2
answers
82
views
Upper rectangle area sum to approximate 1/x between $1\leq x\leq 3$
I am trying to figure out how to use rectangles to approximate the area under the curve $1/x$ on the interval $[1,3]$ using $n$ rectangle that covers the region under the curve as such.
Here is what I ...
- 1,071
2
votes
1
answer
78
views
$\frac{(1+x)^n}{(1-x)^3}=a_{0}+a_{1}x+a_{2}x^2+\cdots$ show that ${a_{0}+\cdots+a_{n-1}=\frac{n(n+2)(n+7)2^{n-4}}{3}}$
$$\displaystyle{\frac{(1+x)^n}{(1-x)^3}=a_{0}+a_{1}x+a_{2}x^2+\cdots}$$, show that $$\displaystyle{a_{0}+\cdots+a_{n-1}=\frac{n(n+2)(n+7)2^{n-4}}{3}}$$
When i gave this problem to my friends they said ...
user1235604
2
votes
1
answer
212
views
Calculate the value $\lim_{n\to \infty}\frac{\sum_{j=1}^n \sum_{k=1}^n k^{1/k^j}}{\sqrt[n]{(\sum_{j=1}^n j!)\sum_{j=1}^n j^n}}$
As in title, I want to calculate the following value $$\lim_{n\to \infty}\frac{\sum_{j=1}^n \sum_{k=1}^n k^{1/k^j}}{\sqrt[n]{(\sum_{j=1}^n j!)\sum_{j=1}^n j^n}}.$$
Here is my attempt:
Since $\sum_{j=...
- 312
0
votes
1
answer
75
views
where is the mistake in my calculations of $\displaystyle \lim_{n \to \infty} \sum\limits_{k=1 }^n \frac{a_k}{(n+1-k)(n+2-k)}= \lim_{n\to \infty}a_n$
if $\lim\limits_{n \to \infty}a_n =a$ prove that $\displaystyle \lim_{n \to \infty} \sum\limits_{k=1 }^n \frac{a_k}{(n+1-k)(n+2-k)}= a$
define $b_{n-1}= a_n - a_{n-1}$ then $\lim\limits_{n \to \...
- 6,563
3
votes
2
answers
296
views
if $\lim\limits_{n \to \infty} b_n =0 $ then how to prove that $\lim\limits_{n \to \infty} \sum\limits_{k =1} ^n \frac{b_k}{n+1-k}=0$
in Problems in Mathematical Analysis I problem 2.3.16 a),
if $\lim\limits_{n \to \infty}a_n =a$, then find $\lim\limits_{n \to \infty} \sum\limits_{k=1 }^n \frac{a_k}{(n+1-k)(n+2-k)}$
The proof that ...
- 6,563
1
vote
1
answer
230
views
Compute $\lim\limits_{n\rightarrow+\infty}(\sum\limits_{i=1}^n(1+\frac{i}{n})^i)^{\frac{1}{n}}$
Here is a question in calculus. Compute the limit of the sequence: $\lim\limits_{n\rightarrow+\infty}(\sum\limits_{i=1}^n(1+\frac{i}{n})^i)^{\frac{1}{n}}$?
There are in general three ways to compute ...
- 825
11
votes
3
answers
451
views
How to evaluate $ \sum\limits_{k=0} ^{\infty} \frac{(-1)^k}{4k+3}$?
I was trying to solve the integral $\int_0 ^{\frac{\pi}{4}} \sqrt{\tan{x}}dx$ and I noticed I can do the following:
$$\int_0 ^{\frac{\pi}{4}} \sqrt{\tan{x}}dx=\int_0 ^{\frac{\pi}{4}} \sqrt{\tan{x}} \...
- 6,563
0
votes
2
answers
160
views
$\lim_{{n \to \infty}} \sum_{{k=1}}^{n} \arctan\left(\frac{1}{k}\right) - \ln n$
$\arctan(x) = \sum_{m=0}^{\infty} \frac{(-1)^m}{2m+1}x^{2m+1}$
\begin{align*}
\arctan\left(\frac{1}{k}\right) &= \sum_{m=0}^{\infty} \frac{(-1)^m}{2m+1}\left(\frac{1}{k^{2m+1}}\right)
&= \frac{...
3
votes
2
answers
169
views
Prove that $\lim_{n\rightarrow\infty}\frac{f(n)}{n!}=e$
Prove that $$\lim_{n\rightarrow\infty}\frac{f(n+1)}{n!}=e\tag{1}$$where $$f(n+1)=n(1+f(n))$$
The recurrence relation of $n!$ is $a_n=na_{n-1}$ or $a_{n+1}=(n+1)a_n$. I thought of making a new ...
- 6,549