All Questions
10
questions
0
votes
1
answer
59
views
$\sum_{n=0}^{\infty} {n \choose y} p^{y+1}(1-p)^{2n-y}$
I am stuck in finding the sum of $\sum_{n=0}^{\infty} {n \choose y} p^{y+1}(1-p)^{2n-y}$.
The sum looks quite similar to a negative binomial sum but I can't really find the exact form. Can anyone help?...
- 243
0
votes
2
answers
69
views
Infinite sum power series
I would like to show
$$
\sum_{r=0}^{\infty}\frac{1}{6^r} \binom{2r}{r}= \sqrt{3}
$$
I have tried proving this using telescoping sum, limit of a sum, and some combinatorial properties but I couldn't ...
- 41
1
vote
1
answer
89
views
Evaluating $\sum^n_{x=1}{2x-1\choose x}t^x$
Is there any technique that I can use to evaluate
$$\sum^n_{k=1}{2k-1\choose k}t^k, \quad \forall t\in\left(0,\frac{1}{4}\right)$$
It can be shown that the series converges even if $n\to \infty$ ...
- 69
10
votes
2
answers
2k
views
Find the value $\binom {n}{0} + \binom{n}{4} + \binom{n}{8} + \cdots $, where $n$ is a positive integer.
Given
$$(1+x)^n= \binom {n}{0} + \binom{n}{1} x+ \binom{n}{2} x^2+ \cdots + \binom {n}{n} x^n.$$
Find the value $\binom {n}{0} + \binom{n}{4} + \binom{n}{8} + \cdots $, where $n$ is a positive ...
- 8,892
4
votes
3
answers
466
views
Series with Binomial Coefficients
I need to get a closed form for this series $$\sum_{x=0}^{\infty} x {z \choose x} \lambda ^ x \mu^{z-x}$$
I know that that $\sum_{x=0}^{\infty} {z \choose x} \lambda ^ x \mu^{z-x} = (\lambda + \mu)^z$...
- 2,342
3
votes
1
answer
224
views
Closed form of the sum of the product of three binomial coefficients
I encountered with this kind of series from the calculation in quantum optics:
$$\sum_{n,m=0}^\infty \sum_{k,l=0}^{\min(n,m)}\binom{n}{k}\binom{m}{l}\binom{n+m-k-l}{m-k}A^{n+m}B^kC^l$$
Provided that ...
- 1,110
4
votes
3
answers
159
views
Find the sum of the $\sum_{m=k}^{+\infty}\binom{m}{k}(1-p)^k\cdot p^{m-k}$
Let $0<p<1$,Find the sum
$$\sum_{m=k}^{+\infty}\binom{m}{k}(1-p)^k\cdot p^{m-k}$$
- 93.6k
1
vote
3
answers
197
views
Find the radius of convergence of the series $y=\sum_{n=0}^{\infty}\binom{p}{n} x^{n}$
Let $p\in R$ Find the radius of convergence of the series:
$$y=\sum_{n=0}^{\infty}\binom{p}{n} x^{n}$$
Show that y satisfies the differential equation $(1+x)y'=py$ and initial condition $y(0)=1$...
- 359
3
votes
0
answers
120
views
Finding a closed form for this summation
I have been trying to derive a few identities using some bell polynomials and a technique i have come up with and i came across this summation:
$$
\rho(n,k) = \sum_{j=0}^k {k \choose j} {\frac{-j}{2} ...
- 1,957
7
votes
2
answers
731
views
Prove $(1-x)^{2k+1} \sum\limits_{n\ge 0}\binom{n+k-1}{k}\binom{n+k}{k} x^n = {\sum\limits_{j\ge 0} \binom{k-1}{j-1}\binom{k+1}{j} x^j} $
I stumbled upon the identity
$$(1-x)^{2k+1} \sum\limits_{n\ge 0}\binom{n+k-1}{k}\binom{n+k}{k} x^n = {\sum\limits_{j\ge 0} \binom{k-1}{j-1}\binom{k+1}{j} x^j}. $$
The right-hand side is a polynomial. ...
- 1,359