All Questions
1,809
questions
-1
votes
1
answer
46
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Resources to master summation symbol [closed]
I noticed that I have some difficulties to use the summation tools( change of index, double or multiples summation...). Do you have some resources or book to master this topic. I am using concrete ...
- 9
1
vote
0
answers
68
views
Evaluating an infinite series with a function
There is an infinite series, I want to transform it into a function, with digamma functions or something else. I hope someone can provide some guidance and suggestions.
$$
E(x,y)=\sum_{n=-\infty}^{\...
- 21
6
votes
2
answers
251
views
Problematic limit $\epsilon \to 0 $ for combination of hypergeometric ${_2}F_2$ functions
In an earlier question, the integral $$I_n(c)=\int_0^\infty x^n (1+x)^n e^{-n c x^2} dx$$ was considered with particular focus on its behavior for positive integer $n$. In trying to analyze this, it ...
- 16.6k
3
votes
0
answers
48
views
How to solve $\sum\limits_{n=1}^\infty \prod\limits_{k=0}^{n-1} \frac{\alpha+k}{\beta+k}$? [duplicate]
This problem: $S:=\sum\limits_{n=1}^\infty \prod\limits_{k=0}^{n-1} \frac{\alpha+k}{\beta+k}$ where $\beta > a+1, \ \ \alpha, \beta >0$ is in my problem book and I couldn't solve it
I tried to ...
- 6,352
0
votes
1
answer
59
views
Cool identities/properties involving the Alternating Harmonic Numbers
Using the following analytic continuation for the Alternating Harmonic Numbers ($\bar{H}_x=\sum_{i=1}^x\frac{(-1)^{i+1}}i$): $$\bar{H}_x=\ln2+\cos(\pi x)\left(\psi(x)-\psi\left(\frac x2\right)-\frac1x-...
- 6,539
0
votes
2
answers
89
views
A threshold for an exponential sum
I came across a sum where I have to find the smallest $n$ so that
$$\sum_{x = 0}^n \frac{250^x}{x!} \ge \frac{e^{250}}{2}$$
I wrote a Java code and the result was 55 but with Desmos it was over 129 (...
1
vote
0
answers
41
views
A partial sum formula [duplicate]
I'm very familiar with partial sums and such little bit hard once, but I was wondering is there a partial sum formula for that
$$\displaystyle\sum_{n=1}^k n^n$$
I have tried with Wolfram alpha but I ...
- 21
1
vote
0
answers
81
views
Closed form for $ \sum_{a_1=0}^\infty~\sum_{a_2=0}^\infty~\cdots~\sum_{a_n=0}^\infty \dfrac1{(a_1!+a_2!+\ldots+a_n!)} $ [closed]
After reading this post and the general solution for that case, I wonder if there is a closed form for the general solution for this sum:
$ \sum_{a_1=0}^\infty~\sum_{a_2=0}^\infty~\cdots~\sum_{a_n=0}^\...
- 843
0
votes
1
answer
77
views
Please help me to find the sum of an infinite series. [duplicate]
Please help me to solve this problem. I need to find the sum of an infinite series:
$$
S = 1 + 1 + \frac34 + \frac12 + \frac5{16} + \cdots
$$
I tried to imagine this series as a derivative of a ...
- 21
2
votes
2
answers
80
views
Evaluate $\sum_{k=1}^{\infty}\frac{9k-4}{3k(3k-1)(3k-2)}$
We want to evaluate the series: $$\sum_{k=1}^{\infty}\frac{9k-4}{3k(3k-1)(3k-2)}$$
My try :
We have :
$$\frac{9k-4}{3k(3k-1)(3k-2)}=\frac{1}{3k-1}+\frac{1}{3k-2}-\frac{2}{3k}$$
Therefore:
$$\sum_{k=1}^...
- 2,288
1
vote
2
answers
56
views
Summation form of improper integrals
On page 9, Edwards has this expression
$$ \int_0^{\infty} e^{-nx} x^{s-1} dx = \frac{\Pi(s-1)}{n^s}$$
obtained from Euler’s factorial formula by replacing $x$ with $nx$. Can you help with the next ...
- 437
2
votes
2
answers
76
views
How to calculate thi sum $\sum_{n=2}^{\infty} \frac{\left( \zeta(n) - 1 \right) \cos \left( \frac{n \pi}{3} \right)}{n}$
My question
$$ \displaystyle{\mathcal{S} = \sum_{n=2}^{\infty} \frac{\left( \zeta(n) - 1 \right) \cos \left( \frac{n \pi}{3} \right)}{n}}$$
My try to solve the integral
$$\displaystyle \sum\limits_{n =...
- 121
0
votes
0
answers
14
views
Getting the formular of a summation [duplicate]
im kind of stuck at my math homework from my calculus class. To progress further i need to be able to write a Summation into a forumular(?), as seen in the picture. Is there any proven method to do ...
- 1
0
votes
1
answer
94
views
Showing $\sum_{n=1}^{\infty }\left ( \sum_{j=1}^{\infty }\frac{x^{(n-j)^2}-x^{(n+j-1)^2}}{(2n-1)(2j-1)} \right ) = \frac{\pi^2}{8}$
Show that
$$\sum_{n=1}^{\infty }\left ( \sum_{j=1}^{\infty }\frac{x^{(n-j)^2}-x^{(n+j-1)^2}}{(2n-1)(2j-1)} \right) = \frac{\pi^2}{8}$$
I liked this problem because the result is a final answer, and ...
- 1,384
1
vote
5
answers
111
views
Alternative ways to evaluate $\sum_{k=1}^{n}(2k+1)^2$
I'm looking for alternative ways to calculate $$\sum_{k=1}^{n}(2k+1)^2$$
The normal approach is to expand $(2k+1)^2$ and use the formulas $\sum_{k=1}^n k^2 = \dfrac{n(n+1)(2n+1)}6$ , $\sum_{k=1}^n k = ...
- 6,790
0
votes
4
answers
195
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.
$$\sum_{n=3}^ \...
- 6,352
0
votes
1
answer
45
views
Sum sequence using Stolz–Cesàro
I have this sequence, and I need to find the convergence of the sum sequence.
The answer is - sum equal π/4.
But I tried to solve it by Stolz–Cesàro, as you can see in the picture, And what I got is ...
- 215
1
vote
1
answer
46
views
Simplifying $\frac{1}{2}\sum_{n=0}^{\infty}{(n+1)(n+2)(z^n+z^{n+1})}$
This is part of a larger problem where I am trying to prove $\sum_{n=0}^{\infty}{(n+1)^2z^n}=(z+1)/(1-z)^3$. Thus far I have used the derivatives of the geometric series to obtain $\frac{1}{2}\sum_{n=...
- 142
4
votes
1
answer
124
views
Find value of this sum
Let $$\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)$$
I think this question is ...
- 4,667
0
votes
2
answers
200
views
how to calculate $\sum\limits_{k=1}^{+\infty }{\arctan \frac{1}{1+k^{2}}}$
Question: how to calculate $$\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}^{...
2
votes
1
answer
176
views
Show that $ \int_0^{\pi\over 2}\frac{\sin(2nx)}{\sin^{2n+2}(x)}\frac{1}{e^{2\pi \cot x}-1}dx =(-1)^{n-1}\frac{2n-1}{4(2n+1)} $
Show that $$ \int_0^{\pi\over 2}\frac{\sin(2nx)}{\sin^{2n+2}(x)}\frac{1}{e^{2\pi \cot x}-1}dx =(-1)^{n-1}\frac{2n-1}{4(2n+1)}
$$
My attempt
Lemma-1
\begin{align*}
\frac{\sin(2nx)}{\sin^{2n}(x)}&=\...
0
votes
0
answers
130
views
Calculation of $\sum_{n=1}^{\infty}\frac{\psi_1(n)}{2^nn^2}$
Calculation of $$\sum_{n=1}^{\infty}\frac{\psi_1(n)}{2^nn^2}$$
My attempt
\begin{align*}
\sum_{n=1}^\infty\frac{\psi_1(n)}{2^n n^2} &= -\sum_{n=1}^\infty\psi_1(n)\left(\frac{\log(2)}{2^n n}+\int_0^...
4
votes
2
answers
196
views
How to evaluate this sum $\sum_{n=1}^{\infty} \frac{(-1)^n}{(n^2 + 3n + 1)(n^2 - 3n + 1)}$
How to evaluate this sum $$\sum_{n=1}^{\infty} \frac{(-1)^n}{(n^2 + 3n + 1)(n^2 - 3n + 1)}$$
My attempt
$$\sum_{n=1}^{\infty} \frac{(-1)^n}{(n^2 + 3n + 1)(n^2 - 3n + 1)}$$
$$= \sum_{n=1}^{\infty} \...
8
votes
2
answers
243
views
How to calculate $\int _0^1 \int _0^1\left(\frac{1}{1-xy} \ln (1-x)\ln (1-y)\right) \,dxdy$
Let us calculate the sum
$$
\displaystyle{\sum_{n=1}^{+\infty}\left(\frac{H_{n}}{n}\right)^2},
$$
where $\displaystyle{H_{n}=1+\frac{1}{2}+\cdots+\frac{1}{n}}$ the $n$-th harmonic number.
My try
The ...
1
vote
1
answer
123
views
Convergence of summation of complex exponentials with alternating exponent
Related to my previous question, consider $$f(s)=\sum_{k=1}^{\infty} \exp(-s(-2)^k)$$where $s\in\mathbb{C}$ is a complex number. According to the Willie Wong's comment, $f(s)$ diverges when $\Re\{s\} \...
- 4,359
2
votes
2
answers
225
views
Evaluate the infinite product $ \prod_{n=1}^{\infty} \left ( 1 + \frac{x^2}{n^2+n-1} \right )$
Question statement
Evaluate the infinite product
$$\displaystyle{\prod_{n=1}^{\infty} \left ( 1 + \frac{x^2}{n^2+n-1} \right ) }$$
My try
Because of the square of $\displaystyle{x}$ , we can consider $...
1
vote
1
answer
129
views
Borel Regularization of $\sum_{n=1}^\infty \ln(n)$ [closed]
I'm trying to solve the following taylor series
$$\sum_{n=0}^\infty \frac{x^n}{n!} \ln(n+1)$$
so I can regularize the following sum
$$\sum_{n=1}^\infty \ln(n)$$
Using Borel Regularizaiton I can use ...
0
votes
1
answer
41
views
How to express $\sum_{i=0}^{m} \exp [(\frac{a}{b+c+i})^2] $ in terms of an integral?
I have this sum
$$\sum_{i=0}^{m} \exp [(\frac{a}{b+c+i})^2] $$
where the upper limit $m$ is a finite non-negative integer, and $a,b,c\in\mathbb{R}$. I want to transform summation to an integral ...
- 51
0
votes
0
answers
33
views
Question on transforming a sum to an integral using the Euler–Maclaurin formula.
I have a question regarding transforming a summation to an integral using the Euler–Maclaurin formula. Imagine I have this sum
$$\sum_{i=0}^{m} f(i) \qquad \text{with} \qquad f(i)= \exp [(\frac{a}{b+...
- 51
1
vote
0
answers
69
views
Leibniz integral rule for summation
Context
Fundamental points of Feymann trick:
You have an integral $I_0=\int_a^b f(t)\mathrm{d}t$
Now consider a general integral $I(\alpha)=\int_a^b g(\alpha,t)\mathrm{d}t$ so that $I'(\alpha)=I_0$ ...
