All Questions
1,809
questions
0
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1
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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 ...
- 267k
-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 ...
- 18.8k
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 ...
- 144k
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.
$$
E(x,y)=\sum_{n=-\infty}^{\...
- 21
1
vote
1
answer
68
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Evaluate $\lim\limits_{n \to \infty} \sum_{k=1}^n \frac{k^p}{n^{p+1}}$ [duplicate]
Let $p$ be a real number. Evaluate $\displaystyle \lim_{n \to \infty} \sum_{k=1}^n \dfrac{k^p}{n^{p+1}}$.
I think this depends on the value of $p$ because then large $n$ would mean small $n^{p+1}$ ...
- 54.2k
3
votes
0
answers
48
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How to solve $\sum\limits_{n=1}^\infty \prod\limits_{k=0}^{n-1} \frac{\alpha+k}{\beta+k}$? [duplicate]
This problem: $S:=\sum\limits_{n=1}^\infty \prod\limits_{k=0}^{n-1} \frac{\alpha+k}{\beta+k}$ where $\beta > a+1, \ \ \alpha, \beta >0$ is in my problem book and I couldn't solve it
I tried to ...
- 6,352
2
votes
1
answer
92
views
Converting an Integral to a Sum
I have an integral over $t$ where I choose $t$ takes discrete values $t=0,1,2,...$ and would like to write the integral as a sum using the same function. I realize that this is not necessarily ...
- 44.1k
0
votes
1
answer
59
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Cool identities/properties involving the Alternating Harmonic Numbers
Using the following analytic continuation for the Alternating Harmonic Numbers ($\bar{H}_x=\sum_{i=1}^x\frac{(-1)^{i+1}}i$): $$\bar{H}_x=\ln2+\cos(\pi x)\left(\psi(x)-\psi\left(\frac x2\right)-\frac1x-...
- 6,539
6
votes
3
answers
318
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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 ...
- 54.2k
12
votes
4
answers
490
views
Evaluating $\int_{-\infty}^\infty\frac1{1+x^2+x^4+\cdots}\ \text{dx}$
I have calculated that
$$\begin{align}
\int_{-\infty}^\infty\frac1{1+x^2}\ \text{dx}&=\pi \\
\int_{-\infty}^\infty\frac1{1+x^2+x^4}\ \text{dx}&=\frac\pi{\sqrt3} \\
\int_{-\infty}^\infty\frac1{...
- 22.1k
0
votes
2
answers
89
views
A threshold for an exponential sum
I came across a sum where I have to find the smallest $n$ so that
$$\sum_{x = 0}^n \frac{250^x}{x!} \ge \frac{e^{250}}{2}$$
I wrote a Java code and the result was 55 but with Desmos it was over 129 (...
- 267k
3
votes
2
answers
76
views
Does $\lim_{n \to \infty}\frac{n}{n + \sum_{k=1}^{n}k}$ converge to $0$ or $1$?
This converges. I am asking whether it converges to $0$ or $1$ because both seem to make sense. Using LH, the sum becomes
$$\displaystyle\lim_{n \to \infty}\frac{n}{n + \displaystyle\sum_{k=1}^{n}k}\...
- 430k
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 ...
- 54.2k
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 ...
- 39.4k
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