All Questions
Tagged with sequences-and-series closed-form
941
questions
2
votes
2
answers
75
views
Is $\sum _{n=0} ^{\infty} \frac{1}{(2n+1)^2(2n+3) \binom{n-1/2}{n}} = G - \frac{1}{2}$?
Wolfram Mathematica claims that
$$\sum _{n=0} ^{\infty} \frac{1}{(2n+1)^2(2n+3) \binom{n-1/2}{n}} = G - 1/2$$
where $G$ is the Catalan constant.
I can't figure out how to prove that. I found a similar ...
4
votes
1
answer
100
views
Sum of reciprocal Bernoulli numbers
What is sum of the Bernoulli numbers? discusses the sum of the Bernoulli numbers, using divergent sum methods since the Bernoulli numbers grow exponentially. This exponential growth makes it so that ...
- 8,945
8
votes
0
answers
277
views
What is $\min\{ m+k\}$ such that $ \binom{m}{k}=n$?
Now asked on here.
Most numbers in pascal triangle appear only once ( excluding the duplicates in the same row of the Pascal's triangle) but certain numbers appear multiple times. For instance, 6 ...
- 6,620
5
votes
5
answers
226
views
Closed formula for probability of n-digit numbers containing three consecutive sixes
I'm trying to find a closed formula $f(n)$ for the probability of choosing a number with $n$ digits that contains at least three consecutive sixes. Ideally, the formula should not depend on $f(n-1)$. ...
-1
votes
0
answers
71
views
Need Help Simplifying a Series Involving Exponential and Factorial Terms
I'm working on solving a fractional differential equation and encountered the following series:
$$
\sum_{n = 1}^{\infty}
\frac{\displaystyle\beta^{n}n^{p}\,{\rm e}^{...
6
votes
1
answer
143
views
Is there a closed form for the quadratic Euler Mascheroni Constant?
Short Version:
I am interested in computing (as a closed form) the limit if it does exist:
$$ \lim_{k \rightarrow \infty} \left[\sum_{a^2+b^2 \le k^2; (a,b) \ne 0} \frac{1}{a^2+b^2} - 2\pi\ln(k) \...
- 17.5k
3
votes
1
answer
128
views
how to evaluate $\int_{0}^{1}\int_{0}^{1} \frac{2-x-y}{\sqrt{xy}(1+xy) \log_{\pi}^{2}{xy}} \, dx \, dy$
How to evaluate: \begin{align*}
&\int_{0}^{1}\int_{0}^{1} \frac{2-x-y}{(\sqrt{xy}+\sqrt{x^{3} y^{3}}) \left[ \log_{\pi}^{2}{x} + 4\log_{\pi}{\sqrt{x}} \log_{\pi}{y} + \log_{\pi}^{2}{y} \right]} \, ...
2
votes
2
answers
138
views
A question about sum with reciprocal quartic
Evaluate
$$
\sum_{n=1}^{\infty} \frac{n+8}{n^{4}+4}
$$
According to WolframAlpha it is $\pi\coth(\pi) - \dfrac{5}{8}$
My attempt:
I tried to separate $\dfrac{n}{n^{4}+4}$ and $\dfrac{8}{n^{4}+4}$. ...
- 192
0
votes
1
answer
32
views
Closed form expression for series $f_\alpha(x) = \sum_{n\geq 0} (-n)^{\alpha} x^{2n}$
Let $\alpha \in \mathbb N$, and consider the Taylor series
$$f_\alpha(x) = \sum_{n >0} (-n)^{\alpha} x^{2n}$$
which is convergent for $|x|<1$.
Question: can we find a closed form to express $...
- 5,920
0
votes
1
answer
116
views
Identify the special function with this sum and integral form
There is a multivariate generalization of the Bessel function that has both a sum and integral form. Both are functions of a vector $\mathbf{0}\leq\mathbf{x}\in\mathbb{R}^n$, with parameters defined ...
- 381
0
votes
1
answer
73
views
closed form of $\sum_{n=1}^\infty \frac{J_{2n-1}(z)}{2n-1}$
Does the following infinite series have a closed form:
\begin{equation}
\sum_{n=1}^\infty \frac{J_{2n-1}(z)}{2n-1}?
\end{equation} Here, $J_n$ is the Bessel function. (If the denominator does not ...
- 159
0
votes
0
answers
118
views
Closed-form solution for series $\sum_{n=1}^{\infty}\frac{a^{n}}{\sqrt{n!}}$ involving square root of a factorial
We have $e^{x}=\sum_{n=0}^{\infty}\frac{x^{n}}{n!}$, however, we are facing
$\sum_{n=1}^{\infty}\frac{a^{n}}{\sqrt{n!}}$ involving square root of a factorial.
Since $n!>n^{2}$ if $n\geq4$, so $\...
- 301
0
votes
3
answers
75
views
How do I calculate the closed form of $\sum_{k=2}^\infty kx^{k-2}$
This is an exercise from Wade, the answer is given as; $$\sum_{k=2}^\infty kx^{k-2}=\frac{2-x}{(1-x)^2},$$ but there is no help as to how to arrive at that answer. I have completed the first question ...
- 3
5
votes
2
answers
105
views
Simplify in closed-form $\sum_n P_n(0)^2 r^n P_n(\cos \theta)$
Simplify in a closed form the sum $$S(r,\theta)=\sum_{n=0}^{\infty} P_n(0)^2 r^n P_n(\cos \theta)$$ where $P_n(x)$ is the Legendre polynomial and $0<r<1$. Note that $P_n(0)= 0$ for odd $n$ and $...
- 1,258
6
votes
2
answers
261
views
Find the closed form of $_3F_2(\frac{1}{4},\frac{3}{4},\frac{5}{4};\frac{3}{2},\frac{7}{4};1)$
Context
Some investigation suggests that the following identity is true:
\begin{align}
_3F_2(\frac{1}{4},\frac{3}{4},\frac{5}{4};\frac{3}{2},\frac{7}{4};1)=\frac{3\sqrt{2}\sqrt{\pi}\left(2\log({1+\...
- 323