5
votes
2answers
64 views

Challenge: How to prove this identity between bi- and trinomial coefficients?

This question is the continuation of its predecessor. Using the convention that trinomial coefficients $$ \binom{n}{k_1,k_2,k_3}=\frac{n!}{k_1! k_2! k_3!} $$ are zero if $k_i<0$ or $\sum_i k_i\neq ...
2
votes
2answers
40 views

Infinite sum of ratio of factorials

Mathematica tells me that for $r, p$ positive integers with $\mathcal{r} < p$ we have $$\sum_{i=p+1}^\infty \frac{(i-r)!}{(i+r)!} = \frac{(p+1-r)!}{(2r-1)(p+r)!}. $$ Can anyone point me in the ...
0
votes
2answers
22 views

Factorial Summation Problem [duplicate]

$$\sum_{j=0}^n j\cdot j!$$ I got $(n+1)!-1$ as the answer but I'm not sure if that's right or how I even got to that answer exactly. (my paper is a mess of random work and I can't make it out). Can ...
1
vote
2answers
39 views

Factorial as a sum. Insight appreciated

I recently posted an answer to a question about ways to express the factorial function as a sum. I posted the following formula, which I discovered several years ago and I haven't seen anywhere else: ...
2
votes
1answer
83 views

Challenge: How to prove this reduction identity for factorials of even numbers?

Some time ago, as a by-product of a proof, I came across an odd (at least to me) identity for reducing the factorial of an even number into a sum: $$(2n)!=\sum_{k=0}^{\lfloor \frac{n}2 \rfloor} ...
1
vote
2answers
47 views

Using induction to prove that $\sum_{r=1}^n r\cdot r! =(n+1)! -1$

Use induction to prove that $\displaystyle\sum_{r=1}^n r\cdot r! =(n+1)! -1$ I first showed that the formula holds true for $n=1$. Then I put n as $k$ and got an expression for the sum in ...
8
votes
8answers
166 views

Telescoping series of form $\sum (n+1)\cdot…\cdot(n+k)$

Wolfram Alpha is able to telescope sums of the form $\sum (n+1)\cdot...\cdot(n+k)$ e.g. $(1\cdot2\cdot3) + (2\cdot3\cdot4) + (...) + n(n+1)(n+2)$ How does it do it? EDIT: We can rewrite as: $\sum ...
2
votes
2answers
69 views

Why does this sum equal to (4^n -1)

How do I get to this solution? $\sum _{k=1}^n\left(\binom nk 3^{n-k}\right)=\left(4^n-1\right)$ I believe it's connected to this, which I know is true: $\sum \:_{k=1}^n\binom nk=2^n-1$
1
vote
0answers
118 views

The closed form of $\sum_{n=1}^{x}n!$

Let $$y=\sum_{n=1}^{x}n!$$ be the sum of consecutive factorials. What is closed form for $y$ in terms of $x$? Wolfram Alpha says that $$y=-(-1)^x\Gamma(x+2)(!(-x-2))-!(-1)-1$$ where $!x$ is ...
7
votes
4answers
161 views

A closed form of $\sum_{k=0}^\infty\frac{(-1)^{k+1}}{k!}\Gamma^2\left(\frac{k}{2}\right)$

I am looking for a closed form of the following series \begin{equation} \mathcal{I}=\sum_{k=1}^\infty\frac{(-1)^{k+1}}{k!}\Gamma^2\left(\frac{k}{2}\right) \end{equation} I have no idea how to ...
3
votes
4answers
128 views

Proving Combinatorical Summation: $n!=\sum_{k=0}^n(-1)^k\binom{n}{k}(n-k)^n$ [duplicate]

been stuck with this question for the last few hours, any help would be appreciated. $$ {\large n! = \sum_{k = 0}^{n}\left(-1\right)^{k}{\,n\, \choose \,k\,} \left(\,n - k\,\right)^{n}} $$ what I ...
3
votes
0answers
58 views

Question about factorial function [duplicate]

Show that $$n!=1+\left(1āˆ’{1 \over 1!}\right)n+\left(1āˆ’{1 \over 1!}+ {1 \over 2!}\right)n(nāˆ’1)+\cdots$$ I can't figure out how this can be solved . I tried to use binomial theorem but I couldn't prove ...
4
votes
3answers
100 views

Factorial identity $n!=1+(1-1/1!)n+(1-1/1!+1/2!)n(n-1)+\cdots$

Show that $\displaystyle{n!=1+\left(1-\frac1{1!}\right)n+\left(1-\frac1{1!}+\frac1{2!}\right)n(n-1)+\cdots}$. I can't figure out how this can be solved. I tried to use the binomial theorem but I ...
3
votes
3answers
95 views

The sum $1!+2!+3!+…+2007!$ is not a perfect square

Today my teacher told me to prove this:- Prove that $1!+2!+3!+...+2007!=\sum_{n=1}^{2007}(n!) $ is neither a perfect square nor a perfect cube. Not getting any idea. Please help. Thanks in advance.
1
vote
1answer
211 views

Factorials in Sigma Notation

Geometric Series one would use $S_n = \dfrac{a_1\cdot (1 - r^n)}{(1 -r)}$. Arithmetic Series one would use $S_n = \dfrac{n\cdot (a_1 + a_n)}{2}$. But how would I convert a sigma notation problem with ...
0
votes
2answers
29 views

How can we prove this = 1 for all n

$\displaystyle n!-\sum_{k=1}^{n-1}k\cdot k!$ By computing this by hand for several small values of $n$ I can see that it is always equal to 1. But I can't see how to prove that.
2
votes
0answers
58 views

How to prove these indentities? [closed]

How to prove these indentities? $\displaystyle \sum \limits_{k\geq0} {2n\choose 2k-1}{k-1\choose m-1}=2^{2n-2m+1}{2n-m\choose m-1}$ $\displaystyle \sum \limits_{k=0}^{m} {m\choose k}{n+k\choose ...
2
votes
1answer
50 views

Calculate sum wtih binomial coefficients

I need help with finding the sum of $\sum \limits_{k=0}^{n} \frac{1}{k+1}{n\choose k}x^{k+1}$
0
votes
2answers
45 views

How to calculate this sum

How do you calculate this sum $ \sum \limits_{k=1}^{n} \frac{k}{n^k}{n\choose k}$ ?
0
votes
7answers
152 views

Calculating $\displaystyle\sum_{i=1}^{n} \binom{i}{2}$

Show $\displaystyle\sum_{i=1}^{n} \binom{i}{2}=\binom{n+1}{3}$. I'm thinking right now (though not getting anywhere with it) that I want to expand out the summation portion to $i!/2!(i-2)!$ and ...
9
votes
2answers
162 views

Prove $\sum_{n=1}^\infty(e-\sum_{k=0}^n\frac1{k!})=1$

This comes from the comments section of this question here, credits Lucian. The statement is $$\sum_{n=1}^\infty\left(e-\sum_{k=0}^n\frac1{k!}\right)=1$$ This looks really interesting, so I was ...
0
votes
1answer
92 views

Digit in units place of 1!+2!+…99!

There isn't much I can add to the question description to expand upon the title. I came across this in a multiple choice test. The options were 3, 0, 1 and 7. I am absolutely stumped. Any pointers? By ...
9
votes
4answers
353 views

Evaluate a finite sum with four factorials

Given positive integers $k, m, n$ such that $1 \leq k \leq m \leq n$. Evaluate $$ \sum^{n}_{i\mathop{=}0}\frac{1}{n+k+i}\cdot\frac{(m+n+i)!}{i!(n-i)!(m+i)!}$$ Any hints? I'm stuck on ...
3
votes
2answers
121 views

Why does $\sum_{i=0}^{\infty}\frac{a^i}{i!}=e^a$?

Here is a standard identity: $$\sum_{i=0}^{\infty}\frac{a^i}{i!}=e^a$$ Why does it hold true?
1
vote
2answers
88 views

Solving an infinite summation involving exponential and factorial

I'm trying to understand an equality that I found in this biology article. $$\sum_{i=0}^\infty\frac{e^{-x}x^i(1-y)^i}{i!} = e^{-x\cdot y}$$ Can you help me proving this equation holds true?
0
votes
1answer
40 views

Summation of a curious series-repeated division by primes

I am interested in knowing if there is some closed form/formula for the following series: ...
3
votes
1answer
107 views

Sum of fraction of factorials

Can anybody explain this? $$\sum\limits_{k=1}^{\frac{m-1}2}\frac{(2k)!(2m-2k)!}{(2k-1)(2m-2k-1)k!^2(m-k)!^2}=\frac{(2m)!}{(2 m-1)m!^2}$$ I did actually simplify this to: ...
5
votes
4answers
247 views

factorial as difference of powers: $\sum_{r=0}^{n}\binom{n}{r}(-1)^r(l-r)^n=n!$?

The successive difference of powers of integers leads to factorial of that power. I think this is the formula $\sum_{r=0}^{n}\binom{n}{r}(-1)^r(l-r)^n=n!$ But I found no proof on internet. Please ...
3
votes
2answers
87 views

Prove that $\sum_{k=0}^n\frac{1}{k!}\geq \left(1+\frac{1}{n}\right)^n$ [duplicate]

It basically says it all in the title. I tried solving the inequality using the bernoulli inequality somehow $$\dfrac{\displaystyle\sum_{k=0}^n\frac{1}{k!}}{(1+\frac{1}{n})^n}\geq 1,$$ but the ...
2
votes
4answers
49 views

A question about sums and factorials

Consider the sum $S=x!+\sum_{i=0}^{2013}i!$, where $x$ is a one-digit nonnegative integer. How many possible values of $x$ are there so that S is divisible by 4?
3
votes
1answer
132 views

Asymptotics for sums involving factorials

This question is rather general, but I have recently encountered the following situation in a variety of different settings. Let us suppose that we are given a complicated sum involving factorials ...
3
votes
2answers
44 views

Find the antiderivative of a function with a finite series and factorials

If $n\in\mathbb{N},s\leq n$, I know that $$ \int_0^1 t^s(1-t)^{n-s-1}dt=\frac{s!(n-s-1)!}{n!}. $$ I would like to find a similar formula: is there a function $f(t)$ such that $$ \int_0^1 f(t) ...
6
votes
1answer
920 views

Is the sum of factorials of first $n$ natural numbers ever a perfect cube?

If $S_n = 1! + 2! + 3! + \dots + n!$, is there any term in $S_n$ which is a perfect cube or out of $S_1$, $S_2$, $S_3$, $\dots S_n$ is there any term which is a perfect cube, where $n$ is any natural ...
0
votes
1answer
88 views

Summation of reciprocal of Product of Factorials.

How can this summation be evaluated: $${āˆ‘ {1\over {a_1!a_2!....a_m!}}}$$ Where $$a_1+a_2+.....+a_m=n$$ Also $a_i !=n $ and $m<n$.
1
vote
1answer
47 views

Canceling factorials and exponentials in sum

I'm trying to understand the following the proof. I want to show that $$E\left[\frac{1}{X+1}\right] = \frac{1}{(n+1)p}(1-(1-p)^{n+1})$$ The proof goes like this: $$ \begin{align} ...
0
votes
0answers
65 views

Solving an equation with several summations and a falling factorial

Can you help me to solve quite a complicated inequation that contains summations? I'd like to solve it for b (leaving balone on ...
2
votes
0answers
96 views

Compute $(-1)^n\sum_{k=1}^n (-1)^k\frac{(k+n-1)!}{(k-1)!(k-1)!(n-k)!}$

Compute $(-1)^n\sum_{k=1}^n (-1)^k\frac{(k+n-1)!}{(k-1)!(k-1)!(n-k)!}$ Define $a_{k,m}=\frac{(-1)^{k+m}(n+k-1)!(n+m-1)!}{(k+m-1)[(k-1)!(m-1)!]^2(n-m)!(n-k)!}$ Compute ...
5
votes
4answers
496 views

Find the sum of series $\sum_{n=0}^\infty\frac{(4n)!}{(4n+4)!}$

I wanted to know how can I start to find the sum of the series: $$\sum_{n=0}^\infty\frac{(4n)!}{(4n+4)!}=\frac{1}{4!}+\frac{4!}{8!}+\frac{8!}{12!}\cdots$$ I am having no clue. Thanks.
14
votes
12answers
2k views

Can the factorial function be written as a sum?

I know of the sum of the natural logarithms of the factors of n! , but would like to know if any others exist.
-2
votes
1answer
200 views

Sigma of factorial function? [duplicate]

How would you find this sum mathmatically?$$\sum_{i=0}^\infty \left( \dfrac{1}{2} \right) ^{i!}$$ What techniques would you use to solve this as well?
0
votes
2answers
47 views

Simplyfing Probability equation

I was solving a homework problem, and I had obtained a formula for the required probability in the question. What I wanted to ask could it be more simplified? $$P = \sum_{i=0}^{a}( \frac{a!}{(a-i)!} * ...
3
votes
3answers
349 views

Evaluate $\displaystyle\sum_{k=1}^nk\cdot k!$

I discovered that the summation $\displaystyle\sum_{k=1}^n k\cdot k!$ equals $(n+1)!-1$. But I want a proof. Could anyone give me one please? Don't worry if it uses very advanced math, I can just ...
1
vote
0answers
83 views

Simplifying expression

I am looking for a way to simplify this expression: $$ \sum_{i=0}^{n-k-1} \sum_{j=0}^{k-1} \left[ {n-k-1 \choose i} {k-1 \choose j} ((-1)^{k-1-j} - (-1)^{n-k-1-i}) \times {(n+0.5)! \over ...
1
vote
4answers
554 views

Evaluate a sum involving n choose r

Evaluate: $\sum_{k=0}^6 (-1)^k \binom{6}{k}$ where $\binom{n}{r}= \frac{n!}{r!(n-r)!}$. I'm unsure how to compute the part with $\binom{6}{k}$, it should be something along the lines of ...
1
vote
0answers
107 views

Double summation including power and factorial [duplicate]

I am finding some trouble in computing the following sum: $$\sum_{k=0}^\infty \frac{x^k}{k!}\;\sum_{m=0}^k\frac {y^m}{m!}$$ Could you please provide a result? Thanks in advance
2
votes
2answers
387 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!
5
votes
1answer
238 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
270 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
294 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
295 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 ...