Questions on the factorial function, $n!=n\times(n-1)\times\cdots\times1$. Consider using the tag (gamma-function) if dealing with noninteger arguments.

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4
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1answer
90 views

Proving that $\sum_{k=0}^\infty\frac{2^{-5k}(6k+1)((2k-1)!!)^3}{4(k!)^3} = {1\over\pi}$

While trying to prove that $$(1)\qquad x\sum_{k=0}^\infty\frac{2^{-5k}(6k+1)((2k-1)!!)^3}{4(k!)^3} = 1 \implies x=\pi$$ I got to a point, using W|A, where I have to prove that $$\color{red}{(2)\qquad ...
4
votes
2answers
47 views

Is there a Sum Factorial?

I am curious if there is any addition factorial. Obviously, $$x! = \prod_{n=1}^x n$$ but what I want is a shorthand way of writing: $$\sum_{n=1}^x n$$ So is there such a thing? and if so, what is ...
1
vote
2answers
74 views

How to calculate decimal factorials, like $0.78!$ [duplicate]

When I enter $0.78!$ in Google, it gives me $0.926227306$. I do understand that $n! = 1\cdot2\cdot3 \cdots(n-1)\cdot n$, but not when $n$ is a decimal. I also have seen that $0.5!=\frac12\sqrt{\pi}$. ...
5
votes
0answers
82 views

Primality of $n! +1$

I came across with a problem where I was required to examine primality of $n! +1$ (17! + 1 was the actual number). Although Wilson's Theorem could be manipulated for determining primality of $n! + ...
0
votes
4answers
31 views

Prove $(n!)/n^n \leq 1/2^{k}$, where $k$ is the floor of $n/2$.

I suppose the natural way to prove this is by induction. When I follow the rather natural steps $$\frac{(n+1)!}{(n+1)^{n+1}} = \frac{n!}{(n+1)^{n}} \leq \frac{n!}{(n)^{n}}$$ in order to apply the ...
0
votes
8answers
210 views

Prove that $n! > n^5$

I'm trying to prove that $n! > n^5$ for large enough values of n. While it seems obvious that this should be true, I have no idea how to prove it rigorously. EDIT: So, looking at the comments, ...
3
votes
4answers
84 views

How can I show that $n! \leqslant (\frac{n+1}{2})^n$?

Show that $$n! \leqslant (\frac{n+1}{2})^n \quad \hbox{for all } n \in \mathbb{N}$$ I know that it can be done by induction but I always find line where I do not know what to do next.
0
votes
1answer
30 views

Factorials with fractions

I don't understand how $$ \frac {n(n-1)!+(n+1)n(n-1)!}{n(n-1)!-(n-1)!} $$ becomes $$ \frac {(n-1)![n+(n+1)n]}{(n-1)!(n-1)} $$ and then how it becomes $$ \frac {n+n^2+n}{n-1} $$ I've tried applying ...
0
votes
2answers
82 views

How to evaluate factorials greater than $69!$

How to evaluate factorials greater than $69!$? On my calculator, $69!$ is the largest number I can enter before it gives me a syntax error, most likely due to an overflow. Is there a way to evaluate ...
4
votes
1answer
137 views

L'Hopital's Rule, Factorials, and Derivatives

I have the following limit $\displaystyle\lim_{n\to\infty}\frac{e^n}{n!}$. Now if I try to solve this using this using L'hopital's rule, I won't be able to since I can't take the derivative of $n!$. ...
1
vote
3answers
80 views

Could negative integer factorials be defined in some way?

I know that, calling $F$ such an extension, if we wanted to keep having $$ F(z+1)=(z+1)F(z),$$ letting $ z=-1$ would lead to the absurdity $ 1=0$. Also, $ \Gamma(z)$ has poles at $ z \in ...
0
votes
3answers
36 views

$\frac{(n/2)!}{n!} = \frac{1}{2^{n/2}(n-1)!!}$?

I was working on a puzzle involving some rather complex probability when I arrived at two very distinct methods with very different ways of calculating the probability of solving the puzzle. The ...
4
votes
2answers
97 views

Find the limit $\,\,\, \lim_{n \to \infty}\Big(\big(1+\frac{1}{n}\big)\big(1+\frac{2}{n}\big) \cdots\big(1+\frac{n}{n}\big)\Big)^{1/n} $

What is the limit of: $$ \lim_{n \to \infty}\bigg(\Big(1+\frac{1}{n}\Big)\Big(1+\frac{2}{n}\Big) \cdots\Big(1+\frac{n}{n}\Big)\bigg)^{1/n}? $$ By computer, I guess the limit is equal to ...
1
vote
0answers
49 views

Simple finite series with reciprocal factorials

I'm trying to find the following sum: $$ \sum_{k=0}^n{1\over(n-k)!}{x^{k+2}\over k+2}. $$ The most obvious way is to differentiate wrt to $x$ leading to $$ ...
0
votes
1answer
45 views

Is it possible to find $n-1$ consecutive composite integers

Given an integer $n\geq 2$ ,can we always find an integer $m$ such that each of the $n-1$ consecutive integers $m+2,m+3,.....,m+n$ are composite?
1
vote
2answers
112 views

Factorial Rational Limit

Anything besides the squeeze theorem. Here it is: $$\lim_{n\to\infty} \frac{(2n - 1)!}{{2n}^{n}}$$ Can someone start me off?
2
votes
1answer
30 views

Finding Factorial using Integral Definition

$n! = \int_{0}^{\infty} {e}^{-x}{x}^{n} \,dx$ How can we find $400!$? $400! = \int_{0}^{\infty} {e}^{-x}{x}^{400} \,dx$ Integration by parts is way too complicated, what are the other options?
1
vote
1answer
91 views

A limit with $((n-1)!)^{1/(n-1)}$ and other roots of factorials

How to prove that the following limit is positive? $$ \lim_{n \to \infty}\left(((n-1)!)^{1/(n-1)}-2\left(\frac{((n-1)!)^3}{(2n-2)!}\right)^{1/(n-1)}\right) >0,$$ where $ n\in \mathbb Z, n>1 ...
0
votes
0answers
29 views

Bounds on constant for Stirling approximation

Stirling's approximation says that $$n!\sim\sqrt{2\pi n}\left(\dfrac{n}{e}\right)^n.$$ What is known about constants $c_1$ and $c_2$ such that $$c_1\sqrt{n}\left(\dfrac{n}{e}\right)^n\le n!\le ...
5
votes
1answer
94 views

Show that $n!^{n+1}$ divides $(n^2)!$

My attempt so far is by induction. Let $f(n) = \frac{(n^2)!}{n!^{n+1}}$, I will try showing that $f(n)$ is a positive integer for all $n$. We have $f(0) = \frac{0!}{0!^{n+1}} = 1$. Now assume for ...
0
votes
0answers
43 views

Nearest factorial given a number.

Hi suppose I have given a number lets say 344545.Is there a way to determine he nearest smallest factorial? This is a Multiple choice question(one question 4 option).So what can be the fastest ...
2
votes
1answer
27 views

Calculating the power of prime in factorial by changing base

The greatest power $k$ of a prime $p$ in the prime factorization of $n!$ is equal to $\frac1{p-1}(n-s(n)_p)$, where $s(n)_p$ is the sum of digits of $n$ when represented in base $p$. How to ...
2
votes
4answers
80 views

Closed Form for Factorial Sum

I came across this question in some extracurricular problem sets my professor gave me: what is the closed form notation for the following sum: $$S_n = 1\cdot1!+2\cdot2!+ ...+n \cdot n!$$ I tried ...
1
vote
1answer
27 views

Trouble understanding factorial algebra

I am having trouble understanding some of the algebraic concepts used here. In fact, the entire thing to me makes sense, except for the second red line. I don't understand how the diagonal swap ...
1
vote
1answer
34 views

Integration by parts with Legendre Functions

I need help deriving $\int_{-l}^l [P_l^m(x)]^2 = \frac{2}{2l+1} \frac{(l+m)!}{(l-m)!}$ for the associated Legendre functions I am supposed to use $P_l^m(x) = (-1)^{-m}\int_{-l}^l ...
0
votes
1answer
134 views

COmbinatoric : Guess who is the winner candidate?

National Radio Broadcast will put a contest to guess five winners out of twelve local boxers who will compete to win the best 5 boxers. All twelve boxers are equally good so the chance of winning is ...
9
votes
3answers
1k views

How to define fractional factorials, like 3.6!? [duplicate]

I did not know that you could find an answer for that. However, I can only use Excel so far to do it. How to calculate 3.6! by hand?
1
vote
1answer
56 views

Summing an infinite series

I have been struggling with a problem involving a Markov Chain. To solve it I need to figure out the following ...
0
votes
1answer
39 views

Partial Derangements

There are n people and n houses, such that every person owns exactly one distinct house. Out of these n people, k people are special (k<=n). You have to send every person to exactly one house such ...
1
vote
2answers
48 views

Help with induction proof for recurrent function

I am having issues with the following inductive proof. Prove by induction on $n$ that $$ a(n) = n!\bigg(\frac{1}{0!} + \frac{1}{1!} + \cdots + \frac{1}{(n-1)!}\bigg)$$ for all $n \geq 1,$ where ...
0
votes
0answers
15 views

Query associated with Factorials and Series

I'm struggling to see how we go from Equation 1 to 2 to 3, which are seen below: Equation 1: $$P(Z=k) = p \binom{k/\delta-1}{n-1}(\lambda\delta)^{n}(1-\lambda\delta)^{k/\delta - n} $$ Equation 2: ...
1
vote
2answers
146 views

Summation of Binomial Theorem

The binomial theorem formula: $$\sum\limits_{k=0}^{n} {n \choose k} = \sum\limits_{k=0}^{n}\frac{n!}{k!(n-k)!} = \sum\limits_{k=0}^{n}\frac{n(n-1)(n-2) \cdots (n-k+1)}{k(k-1) \cdots 2\cdot1}.$$ I am ...
0
votes
2answers
34 views

Factorials and Mathematical induction

I'm having a bit of trouble understanding mathematical induction, particularly when there's a question with powers or factorials. For example I have a problem 1 x 1! +2 x 2! + 3 x 3! +... + n x n! = ...
3
votes
1answer
87 views

How to find asymptotics of this sum

Is there any way to find $f(n)$ in this term: $$\sum_{k=2}^n \frac1{\ln \ln(k!^{k!})} \sim f(n)?$$ The tilde symbol means that $$\lim_{n\to∞} \frac{f(n)}{\sum_{k=2}^n \frac1{\ln \ln(k!^{k!})}} = 1$$ ...
0
votes
1answer
19 views

n! mod c where c is a composite number

I am trying to write a program to calculate what is $n! \, \text{mod} \, c$, where $c$ is a composite number. While I understand $a b \, \text{mod} \, c$ is equal to $((a \, \text{mod} \, c) (b \, ...
17
votes
1answer
774 views

Can a double-factorial be a perfect square?

The title says it, basically. My question is $-$ for $ n \ge 2 $, can $n!!$ be a perfect square, where $!!$ represents the double-factorial? My conjecture is no, but I can't seem to be able to find a ...
1
vote
1answer
73 views

Finding the asymptotics of $\sum_{k=1}^n a^k k!$? Note that $a > 0$.

There's no way to use integration method in this case. I also tried to use Stolz–Cesàro theorem, but couldn't find right $y_n$. What method should I use?
1
vote
0answers
22 views

How are the Stirling-based bounds for the factorial function proven?

According to (26) on wolfram mathworld, one has $$\sqrt{2\hspace{-0.04 in}\cdot \hspace{-0.04 in}\pi} \cdot n^{n+(1/2)} \cdot \operatorname{exp}((-n)\hspace{-0.02 in}+\hspace{-0.02 ...
0
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0answers
17 views

factorial division word problem

I'm having some problems understanding the following: There are 3 programs being observed 4 times (total of 12 observations). There are 12 people used to investigate these programs, such that they ...
3
votes
2answers
44 views

Factorials Algebra

I have the following inequality $$5*10^{-10} \geq \dfrac{2^{n+1}}{(n+1)!}$$ Is there any way this can be solved algebraically? If not, is there a method that is better than guessing, for finding the ...
0
votes
1answer
35 views

How many straight lines can be drawn between five points (A, B, C, D, and E), no three of which are colinear?

How many straight lines can be drawn between five points (A, B, C, D, and E), no three of which are colinear? Attempt: Given 5 points, a line consist always of 2 points. Thus the total number of ...
0
votes
1answer
30 views

How many zip codes are as large as 6000-0000, are even numbers, and have a 7 as their third digit?

When they were first introduced, postal zip codes were five digit numbers, theoretically ranging from $00000$ to $99999$. (In reality, the lowest zip code was $00601$ for San Juan, Puerto Rico; the ...
0
votes
3answers
65 views

In how many ways can the word ELEEMOSYNARY be arranged.

In how many ways can be the letters of the word ELEEMOSYNARY be arranged so that the S is always immediately followed by a Y? Attempt: There are 3 Es, and 2 Ys, and and then all letters appear once ...
1
vote
1answer
31 views

How many different eight note melodies within a single octave can be written if black/white keys alternate.

An octave contains 12 distinct notes(on a piano, five black keys and seven white keys). How many different eight notes melodies within a single octave can be written if the black keys and white keys ...
0
votes
2answers
35 views

A coke hand in bridge from deck of cards.

A coke hand in bridge is one where none of the thirteen cards is an an ace or is higher than a 9. What is the probability of being dealt such a hand? Attempt: Suppose the thirteens cards are amoung ...
0
votes
1answer
39 views

Five cards selected out of 52 cards. Find probalbilty sum of the faces is 48 or more.

Five cards are dealt from a standard 52 card deck. What is the probability that the sum of the faces on the five cards is 48 or more? Attempt: Five cards can be selected out of 52 cards by 52_C_5 ...
1
vote
1answer
55 views

What are chances that not all S's will be adjacent given a phrase at random.

IF the letters in the phrase A ROLLING STONE GATHERS NO MOSS are arranged at random, what are the chances that not all the S's will be adjacent. Attempt: Given there are 6 letters that appear twice, ...
1
vote
1answer
113 views

A bridge hand (13 cards) is dealt from a standard 52 card deck. Given events A and B, find $P(A \cup B)$.

A bridge hand (13 cards) is dealt from a standard 52 card deck. Let A be the event that the hand contains four aces. Let B be the event that the hand contains four kings. Find $P(A \cup B)$. Attempt: ...
1
vote
3answers
30 views

Why is $1 \times 3 \times 5 \times \cdots \times (2k-3) = \frac{(2k-2)!}{2^{(k-1)}(k-1)!}$

In order to find out the Catalan numbers from their generating function you have to evaluate the product above. Here is what I thought: \begin{align*} 1 \times 3 \times 5 \times...\times (2k-3) ...
1
vote
1answer
59 views

How many ways different sets of values can be chosen for the $x_s$ , if $x_1 + x_2 + x_3 = 20$?

Your statistics teacher announces a twenty-page reading assignment on Monday that is to be finished by Thursday morning. You intend to read the first $x_1$ pages Monday, the next $x_2$ pages ...