All Questions
1,811
questions
-1
votes
1
answer
46
views
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
votes
0
answers
54
views
Closed form solution for $\displaystyle \sum _{i=r+1}^{k}\frac{1}{i-1} \cdot \frac{( n-i) !}{( k-i) !}$, where $n,k,r$ are constants and $r \leq k<n$ [closed]
Context
I arrived at this summation, while computing a probability of The Secretary Problem.
Question:
Find Closed form for this or if possible simplify
$$
\frac{r}{n}\sum_{i = r + 1}^{k}\frac{1}{i - ...
- 17
1
vote
0
answers
68
views
Evaluating an infinite series with a function
There is an infinite series, I want to transform it into a function, with digamma functions or something else. I hope someone can provide some guidance and suggestions.
$$
E(x,y)=\sum_{n=-\infty}^{\...
- 21
6
votes
2
answers
250
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 ...
- 16.5k
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 ...
- 6,332
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 ...
- 1,362