Questions tagged [factorial]

Questions on the factorial function, $n!=n\cdot(n-1)\cdot...\cdot1$. Consider using the tag (gamma-function) if dealing with noninteger arguments.

The factorial is defined as the product of all positive integers less than or equal to some integer $n$, written $n!$. Multiple $!$'s skip integers, so for example $10!!!=10\cdot7\cdot4\cdot1$.

This function is only defined over integers, but the extends it to all complex numbers that are not non-positive integers.

3322 questions
2
votes
2 answers

How to solve equation with factorial using algebra?

I bring this sample in order to ilustrate $$x! = 2^x + 8$$ I know the answer is $x=4$ but I dunno how to prove it. I mean, if i put the number 4 by observation, tryal and error, I can get the results, but I dunno how to solve it isolating x like…
Luis P.
  • 131
  • 4
2
votes
4 answers

Efficient method to evaluate the following series: $\sum_{n=1}^\infty \frac{n^2\cdot (n+1)^2}{n!}$

How do I calculate the infinite series: $$\frac{1^2\cdot 2^2}{1!}+\frac{2^2\cdot 3^2}{2!}+\dots \quad?$$ I tried to find the nth term $t_n$. $$t_n=\frac{n^2\cdot (n+1)^2}{n!}.$$ So,…
2
votes
3 answers

Divergent sum of factorials

Is it possible to get an exact value of the sum (using divergent series summation methods) $$ \sum_{n=0}^\infty~ \frac{(n+k)!}{n!} \quad?$$ where $k$ is a positive integer. The only other divergent sum of factorials I have seen is…
Matt Majic
  • 363
  • 2
  • 8
2
votes
3 answers

Is $n! \sum_{i=0}^n{\frac{(-1)^i}{i!}}- (n-1)! \bigg[\sum_{i=0}^{n-2}{\frac{(-1)^i}{i!}}+...+\sum_{i=0}^{2}{\frac{(-1)^i}{i!}}\bigg]=(n-1)!$ true?

I am in the middle of doing a problem and has this sort of expression. I have a feeling that the following equality holds: $$n! \sum_{i=0}^n{\frac{(-1)^i}{i!}}- (n-1)!…
Skipe
  • 158
  • 11
2
votes
3 answers

Solving equations with factorials?

I looked on the internet but couldn't find anything relevant, so I was hoping you could help because I have no clue where to even start with how to solve this equations: x! = 6 Obviously trial and error here could work, but is there any way to do it…
Jacob Lee-Hart
  • 121
  • 1
  • 1
  • 2
2
votes
2 answers

I am stuck on proving $\frac1{2!}+\frac2{3!}+\dots+\frac{n}{(n+1)!}=1-\frac1{(n+1)!}$ by induction, could anyone check my work?

I will skip the Base Case step. This is the questions. Use mathematical induction to prove that$$\frac{1}{2!}+\frac{2}{3!}+\cdots+\frac{n}{(n+1)!}=1-\frac{1}{(n+1)!}$$for all integers $n\ge 1$. This is my proof: $$\sum_{i=1}^n \frac{i}{(i+1)!} =…
2
votes
1 answer

Factorials and equivalency

I am not sure if this would be a proper title because I am a bit confused, but I was reading about proving Pascal's Triangle, and there was a proof on here I was following everything that was happening until the poster mentioned $k! = \prod_{j=1}^k…
Jude
  • 319
  • 4
  • 11
2
votes
2 answers

Does it make sense to multiply probabilities?

I got this interesting sum which seems to involve values of the derangement problem: $$ \sum _{n=0}^{\infty } \frac{1}{(2 n+2) (2 n)!}=\frac{e-1}{e}=1-\frac{1}{e},$$ where $1-\frac{1}{e}$ is the probability that some man gets his own hat back [OEIS…
Fred Kline
  • 1,239
  • 2
  • 16
  • 44
2
votes
1 answer

Minimize $x$ in factorial division

My question is that how can we find the smallest natural number, $n$, such that some other number, $x$, divides $n!$. What I mean is that what minimum $n$ such that $x\mid n!$ for $x,n\in \mathbb N$. My thoughts on this were that, we know that when…
user318919
  • 21
  • 1
2
votes
2 answers

Simplify and find $\lim_{n\to \infty}\frac{(2n-1)!}{(2n+1)!}$

So I was calculating $$\lim_{n\to \infty}\frac{(2n-1)!}{(2n+1)!}$$ and couldn't solve it, so I saw the answer sheet and it said that the limit was $0$, I checked the process and they simplified the expression to $$\lim_{n\to\infty}…
ravelinx
  • 211
  • 1
  • 5
2
votes
5 answers

Does the sequence $a^n/n!$ converge?

The sequence when plotted converges to zero because a factorial grows faster than the numerator, but I can not prove that this sequence actually converges.
Joshua Salazar
  • 501
  • 3
  • 17
2
votes
3 answers

Different seating arrangements on a 6 seater

Six actors sit in a row to have their photographs taken. Romeo and Juliet insist on sitting next to each other. Caesar refuses to sit next to Brutus. Falsta and Puck don't mind where they sit. How many completely different ways altogether, can they…
Ernest
  • 21
  • 1
2
votes
1 answer

Binomial Distribution and Proof Relating to Factorials

I am studying probability and statistics at my university but haven't had a solid math course in awhile(mostly forget algebra dealing with factorials)thus I am stuck with the following proof. According to my book there is a recurrence relation…
JmanxC
  • 161
  • 4
2
votes
1 answer

Is there any better way to find n! without just multiplying all the numbers till n?

I was solving some permutation, combination problems where we know that we are to use factorial. So I was thinking if there was any shorter way, maybe a formula, than multiplying all the numbers till $n$ to find $n$! I know how to add series like…
Farhan Fuad
  • 129
  • 7
2
votes
1 answer

Does the series converge or diverge? $\sum_{n=1}^\infty\frac{4^n+n}{n!}$

Does the following series converge or diverge? $$\sum_{n=1}^\infty\frac{4^n+n}{n!}$$
1 2 3
99
100