All Questions
1,810
questions
0
votes
0
answers
39
views
An inequality involving infinite series
Is
\begin{equation}
\sum_{i=1}^{\infty}\frac{(2i-1)^2}{\left(4(2i-1)^2 x+1\right)^{\sigma}}>\sum_{i=1}^{\infty}\frac{4(i-1)^2}{\left(16(i-1)^2 x+1\right)^{\sigma}},\tag{1}
\end{equation}
for all $x,...
- 98
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 ...
- 268k
-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 ...
- 18.9k
6
votes
2
answers
256
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 ...
- 145k
1
vote
0
answers
69
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.
$$
E(x,y)=\sum_{n=-\infty}^{\...
- 21
1
vote
1
answer
68
views
Evaluate $\lim\limits_{n \to \infty} \sum_{k=1}^n \frac{k^p}{n^{p+1}}$ [duplicate]
Let $p$ be a real number. Evaluate $\displaystyle \lim_{n \to \infty} \sum_{k=1}^n \dfrac{k^p}{n^{p+1}}$.
I think this depends on the value of $p$ because then large $n$ would mean small $n^{p+1}$ ...
- 54.3k
3
votes
0
answers
48
views
How to solve $\sum\limits_{n=1}^\infty \prod\limits_{k=0}^{n-1} \frac{\alpha+k}{\beta+k}$? [duplicate]
This problem: $S:=\sum\limits_{n=1}^\infty \prod\limits_{k=0}^{n-1} \frac{\alpha+k}{\beta+k}$ where $\beta > a+1, \ \ \alpha, \beta >0$ is in my problem book and I couldn't solve it
I tried to ...
- 6,565
2
votes
1
answer
92
views
Converting an Integral to a Sum
I have an integral over $t$ where I choose $t$ takes discrete values $t=0,1,2,...$ and would like to write the integral as a sum using the same function. I realize that this is not necessarily ...
- 43.8k
0
votes
1
answer
60
views
Cool identities/properties involving the Alternating Harmonic Numbers
Using the following analytic continuation for the Alternating Harmonic Numbers ($\bar{H}_x=\sum_{i=1}^x\frac{(-1)^{i+1}}i$): $$\bar{H}_x=\ln2+\cos(\pi x)\left(\psi(x)-\psi\left(\frac x2\right)-\frac1x-...
- 6,549
6
votes
3
answers
318
views
Evaluating $\lim_{n\to\infty}\small\left(\frac{1}{\sqrt{n}\sqrt{n+1}}+\frac{1}{\sqrt{n}\sqrt{n+2}}+\cdots+\frac{1}{\sqrt{n}\sqrt{n+n}}\right)$
Problem: Evaluate
$$
\lim_{n\to\infty}\left(\frac{1}{\sqrt{n}\sqrt{n+1}}+\frac{1}{\sqrt{n}\sqrt{n+2}}+\cdots+\frac{1}{\sqrt{n}\sqrt{n+n}}\right).
$$
I have decent familiarity with limits, but I don't ...
- 54.3k
12
votes
4
answers
490
views
Evaluating $\int_{-\infty}^\infty\frac1{1+x^2+x^4+\cdots}\ \text{dx}$
I have calculated that
$$\begin{align}
\int_{-\infty}^\infty\frac1{1+x^2}\ \text{dx}&=\pi \\
\int_{-\infty}^\infty\frac1{1+x^2+x^4}\ \text{dx}&=\frac\pi{\sqrt3} \\
\int_{-\infty}^\infty\frac1{...
- 22.3k
0
votes
2
answers
89
views
A threshold for an exponential sum
I came across a sum where I have to find the smallest $n$ so that
$$\sum_{x = 0}^n \frac{250^x}{x!} \ge \frac{e^{250}}{2}$$
I wrote a Java code and the result was 55 but with Desmos it was over 129 (...
- 268k
3
votes
2
answers
76
views
Does $\lim_{n \to \infty}\frac{n}{n + \sum_{k=1}^{n}k}$ converge to $0$ or $1$?
This converges. I am asking whether it converges to $0$ or $1$ because both seem to make sense. Using LH, the sum becomes
$$\displaystyle\lim_{n \to \infty}\frac{n}{n + \displaystyle\sum_{k=1}^{n}k}\...
- 431k
1
vote
6
answers
102
views
I try to find sum of $\sum_{i=n}^{2n-1} 3+4i$
I try to find sum of $\sum_{i=n}^{2n-1} 3+4i$
$\sum_{i=n}^{2n-1} 3+ \sum_{i=n}^{2n-1} 4i$ =$3 (2n-1-n+1) + 4 \sum_{i=n}^{2n-1} i $ -> am I right?
if I change the index of $4 \sum_{i=n}^{2n-1} i $ to ...
- 54.3k
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 ...
- 39.7k
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}^\...
- 923
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 ...
- 1,530
-1
votes
1
answer
81
views
$\sum _{k=1}^{\infty }{\frac {\coth(k\pi )}{(k\pi )^{4n-1}}}$ [duplicate]
Show that
$${\displaystyle \sum _{k=1}^{\infty }{\frac {\coth(k\pi )}{(k\pi )^{4n-1}}}=\sum _{k=0}^{2n}(-1)^{k-1}\,{\frac {\zeta (2k)}{\pi ^{2k}}}\,{\frac {\zeta (4n-2k)}{\pi ^{4n-2k}}}\qquad n\in \...
- 33.2k
2
votes
2
answers
80
views
Evaluate $\sum_{k=1}^{\infty}\frac{9k-4}{3k(3k-1)(3k-2)}$
We want to evaluate the series: $$\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}^...
- 268k
1
vote
2
answers
57
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
10
votes
3
answers
614
views
Show that $\sum_{n=1}^{+\infty}\frac{1}{(n\cdot\sinh(n\pi))^2} = \frac{2}{3}\sum_{n=1}^{+\infty}\frac{(-1)^{n-1}}{(2n-1)^2} - \frac{11\pi^2}{180}$
What I do so far
\begin{align*}
\text{Show that} \quad &\sum_{n=1}^{+\infty}\frac{1}{(n\cdot\sinh(n\pi))^2} = \frac{2}{3}\sum_{n=1}^{+\infty}\frac{(-1)^{n-1}}{(2n-1)^2} - \frac{11\pi^2}{180} \\
\...
5
votes
3
answers
430
views
Finishing proof of identity $\sum_{k=b}^{n} \binom{n}{k} \binom{k}{b} = 2^{n-b} \binom{n}{b}$
The identity
$$
\sum_{k=b}^{n} \binom{n}{k} \binom{k}{b} = 2^{n-b} \binom{n}{b}\
$$
is one of a few combinatorial identities I having been trying to prove, and it has taken me way too long. I am ...
- 91k
2
votes
2
answers
78
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
$$\displaystyle \sum\limits_{n =...
- 91k
0
votes
0
answers
14
views
Getting the formular of a summation [duplicate]
im kind of stuck at my math homework from my calculus class. To progress further i need to be able to write a Summation into a forumular(?), as seen in the picture. Is there any proven method to do ...
- 1
1
vote
5
answers
113
views
Alternative ways to evaluate $\sum_{k=1}^{n}(2k+1)^2$
I'm looking for alternative ways to calculate $$\sum_{k=1}^{n}(2k+1)^2$$
The normal approach is to expand $(2k+1)^2$ and use the formulas $\sum_{k=1}^n k^2 = \dfrac{n(n+1)(2n+1)}6$ , $\sum_{k=1}^n k = ...
- 34.2k
0
votes
1
answer
95
views
Showing $\sum_{n=1}^{\infty }\left ( \sum_{j=1}^{\infty }\frac{x^{(n-j)^2}-x^{(n+j-1)^2}}{(2n-1)(2j-1)} \right ) = \frac{\pi^2}{8}$
Show that
$$\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}$$
I liked this problem because the result is a final answer, and ...
- 39.7k
1
vote
1
answer
65
views
Simplification of the sum $\sum_{k=0}^M x^k\binom{M}{k}\binom{r}{k}?$
For any $r\in\mathbb{R}$ and $k\in\mathbb{N}$ let
$$\binom{r}{k}=\frac{r(r-1)(r-2)...(r-k+1)}{k!}$$
be a generalized binomial coefficient.
For $k, M\in\mathbb{N}$ and $r,x\in\mathbb{R}$ is there a ...
- 268k
7
votes
2
answers
349
views
Prove that $\sum_{n=0}^\infty \frac{(-1)^n}{3n+1} = \frac{\pi}{3\sqrt{3}}+\frac{\log 2}{3}$
Prove that $$\sum_{n=0}^\infty \frac{(-1)^n}{3n+1} = \frac{\pi}{3\sqrt{3}}+\frac{\log 2}{3}$$
I tried to look at $$ f_n(x) = \sum_{n=0}^\infty \frac{(-1)^n}{3n+1} x^n $$
And maybe taking it's ...
- 113k
22
votes
3
answers
1k
views
Conjecture: $\sum\limits_{k=1}^nk^m=S_3(n)\times\frac{P_{m-3}(n)}{N_m}$ for odd $m>1 \ ;\ =S_2(n)\times\frac{P_{m-2}'(n)}{N_m}$ for even $m$.
When I was in high school, I was fascinated by $\displaystyle\sum\limits_{k=1}^n k= \frac{n(n+1)}{2}$ so I tried to find the general solution for $\displaystyle\sum\limits_{k=1}^n k^m$ s.t $m \in \...
- 6,565
0
votes
4
answers
196
views
How to evaluate $\sum\limits_{n=3}^ \infty \frac{1}{n \ln(n)(\ln(\ln(n)))^2}$
I saw this problem : Prove that $\sum\limits_{n=3}^ \infty \frac{1}{n \ln(n)(\ln(\ln(n)))^2}$ converges, this is an easy problem could be proved using Cauchy condensation test twice.
$$\sum_{n=3}^ \...
- 40.1k