0
votes
5answers
72 views

Factorial of zero is 1. Why? [duplicate]

Why is the factorial of zero, one. What is the mathematical proof behind it?
0
votes
1answer
21 views

Name of numbers in “to the power of” and factorial calculations

In $4*5=20$ , $4$ and $5$ are multiplicands and $20$ is the product. What are the names / labels of the numbers in the following expressions? $2^3=8$ $4!=24$
3
votes
2answers
95 views

Limit of $\frac {n^n}{n!}$ [duplicate]

I have to prove that $$\lim_{n\to \infty} \frac {n^n} {n!}=\infty$$ I've tried to look for a lower bound that also converges to $\infty$ (I don't know if I'm explainig myself correctly), but I ...
1
vote
4answers
149 views

Which is larger :: $y!$ or $x^y$, for numbers $x,y$.

This is a generalization of this question :: Which is larger? $20!$ or $2^{40}$?. No explicit general solution was presented there and I'm just curious :D Thank-you. Edit :: I want a most-general ...
2
votes
1answer
37 views

Problem finding limit - which function is asymptotically larger

I have a homework question, so please don't answer fully but I would appreciate a push in the right direction. Basically we need to figure out if $n^{n+\frac{1}{2}}e^{-n}$ is larger,smaller, or equal ...
32
votes
14answers
3k views

Which is larger? $20!$ or $2^{40}$?

Someone asked me this question, and it bothers the hell out of me that I can't prove either way. I've sort of come to the conclusion that 20! must be larger, because it has 36 prime factors, some of ...
1
vote
0answers
55 views

Double summation of elementary functions

I am finding some trouble on calculating the following double summation: $ \sum_{k=1}^\infty \frac{b^k}{k!k}\sum_{n=0}^{k-1}\frac{(b*x)^n}{(n-1)!} $ Note that the inside sum gives: ...
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
3answers
113 views

How to find this expression $1000! \mod 3^{300}$

How to find this expression $(1000!\mod 3^{300})$?
4
votes
1answer
95 views

What's the arity of the factorial and exponential operations?

I'm having a conflict with the concept of arity, I've read that the factorial is a unary operation and also that the exponentiation is a binary operation but I feel there's something strange, the ...
2
votes
2answers
666 views

factorial Vs power sequence

What sequence is dominant between $ f(n) = n!$ and $ g(n) = 2^n$ or (or $a^n$) I mean $ f/g -> 0 $ or $infinity$
1
vote
0answers
143 views

How to find the last non-zero digit in ${^n\!P_k} $?

What is the procedure of finding the last non-zero element in ${^n\!P_k}$?
0
votes
0answers
121 views

Is it true that $n> a^2\Rightarrow n!>a^n$, $n\in\mathbb{N}, a\in\mathbb{R}$?

If so, how can it be proven? (I have evaluated it up to $n=25$.) If not, does there exist a $k\in\mathbb{R}$ such as that $n> a^k\Rightarrow n!>a^n$, with $n\in\mathbb{N},a\in\mathbb{R}$? It ...