Tagged Questions
2
votes
2answers
32 views
Simplify summation with factorial and binomial coefficients
I would like to know how to simplify the following summation:
$$\sum_{p=0}^n\quad n!\frac{(2p)!}{(p!)^2}\frac{(2(n-p))!}{((n-p)!)^2}$$
Which rules should I use to simplify it?
Thanks!
0
votes
1answer
72 views
Double summation including Laguerre polynomial, exponential and factorial
I am trying to evaluate the following double sum including the Laguerre polynomial and some elementary functions.
$$
\sum_{N=1}^\infty \frac {\lambda^N} {N!}\frac {1}{N}\sum_{n=1}^{N}L_{n-1}(-\pi ...
3
votes
1answer
102 views
Sum involving the hypergeometric function, power and factorial functions
I am finding some trouble in calculating the following sum involving the hypergeometric function, power and factorial functions.
$$
\sum_{y=1}^\infty \mathrm{e}^z \cdot {}_1F_1\left(1-y;2;-z\right) ...
1
vote
2answers
92 views
Simplify summation of factorials
Hello I guess this equality is true but I don't know how to solve it.
$$\sum_{x=0}^{m(1-\text{sel})} (m-1-x)! (m \cdot \text{sel}) \frac{(m(1-\text{sel}))!}{(m(1-\text{sel})-x)!}(x+1) = ...
7
votes
3answers
169 views
Compact formula for $\sum_k k!$ [duplicate]
Is there any compact formula for:
$$\sum_{k=0}^n k!$$
I've tried to find it using one method for summation, but I was able to receive only compact formula for $\sum_k k! \cdot k = (n+1)!-1$
I've ...
0
votes
1answer
39 views
Handling summations with two variables
If I have a summation with let's say $x=0 \dots 500$ and $y=0\dots1500$
$500 \choose x$ $ 1500 \choose y$ $\dfrac{1}{2^{500}}\dfrac{2^{1500-y}}{3^{1500}}$,
How would I handle the constant? If I ...
2
votes
3answers
196 views
Sum of reciprocals of factorials
Could you help me count this sum:
$$ \sum_{n=1}^{9} \frac{1}{n!} $$
I don't think I can use binomial coefficients.
7
votes
6answers
262 views
simplify summation of factorial (random walk)
I suspect that the expression
$$\sum_{n=0}^N \frac{(N-2n)^2}{n!(N-n)!}$$
simplifies to
$$\frac{2^N}{(N-1)!}$$
But I cannot find the intermediate steps. Can someone give me a hint how I can deduce ...
2
votes
3answers
148 views
When is a factorial of a number equal to its triangular number?
Consider the set of all natural numbers $n$ for which the following proposition is true.
$$\sum_{k=1}^{n} k = \prod_{k=1}^{n} k$$
Here's an example:
$$\sum_{k=1}^{3}k = 1+2+3 = 6 = 1\cdot 2\cdot ...
1
vote
4answers
137 views
Find $\sum_{n=1}^{\infty}\frac{1}{n!}$
Find $$\sum_{n=1}^{\infty}\frac{1}{n!}$$
All of the advice I've seen to compute this sum says to use the ratio test, but this is in a chapter BEFORE the ratio test, so the book wants me to solve ...
2
votes
0answers
79 views
Can we simplify the sum $\sum_{i=0}^k \frac{((i-k)a)^i b^{k-i}}{i!}$?
The problome is rewriten here:
$\sum_{i=0}^k \frac{((i-k)a)^i b^{k-i}}{i!}$
where $0<a<1$, k is an integer larger than 1.
I came to this equation when i try to find some probability. I ...
3
votes
1answer
72 views
Factorial Identity - True or False?
Let $x$ and $y$ be positive integers.
Then, is
\begin{align}
\frac{x^{xy}}{(xy)!} = \sum_{k_1+...+k_x = xy} \frac{1}{(k_1)!...(k_x)!}
\end{align}
true, where $k_1$, ..., $k_x$ are all positive ...
