8
votes
3answers
122 views

Showing $n!<e(\frac{n}{2})^n$

I'd like to prove that $n!<e(\frac{n}{2})^n$. What I have so far: $\sqrt[n]{n!} = \sqrt[n]{1\cdot 2 \cdot \ldots \cdot n} \leq \frac{1+\ldots +n}{n}=\frac{(n+1)n}{2n}=\frac{(n+1)}{2}$. Thus ...
-1
votes
1answer
72 views

Relation/connection between $n!$ or $e$ and $2^n$

What is the relation/connection between $n!$ or $e$ and $2^n$ ? Is the there a relation/connection between $n!$ or $e$ and $2^n$?
9
votes
2answers
160 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 ...
3
votes
0answers
59 views

Which series of numbers effectively translates the factorial to the exponential function?

We have the relation of the Bernoulli numbers $$B_{2n} = (-1)^{n+1}\frac {2(2n)!} {(2\pi)^{2n}} \left(1+\frac{1}{2^{2n}}+\frac{1}{3^{2n}}+\cdots\;\right).$$ For $n>1$, the right hand sum ...
3
votes
2answers
96 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 ...
2
votes
4answers
499 views

Which has a higher order of growth, n! or n^n? [duplicate]

In our algorithms class, my professor insists that n! has a higher order of growth than n^n. This doesn't make sense to me, when I work through what each expression means. ...
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
80 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?
-1
votes
2answers
53 views

Showing $(n+1)^n<e^nn!$ by induction

Show $(n+1)^n<e^nn!$ I know why that would be the case using general knowledge and a bit of substitution but am clueless on how to prove it.
6
votes
3answers
125 views

How to prove $(\frac{n+1}{e})^n<n!<e(\frac{n+1}{e})^{n+1}$ without integrating method?

How to prove $$\left(\frac{n+1}{e}\right)^n<n!<e\left(\frac{n+1}{e}\right)^{n+1}$$ without integrating method? In fact we could prove this by noticing that $$i<x<i+1\Rightarrow \ln ...
1
vote
2answers
108 views

How to show the double factorial isn't a polynomial

$(2n-1)!! = \dfrac{(2n)!}{2^{n} \times n!}$ I was wondering how you prove the double factorial is exponential. I guess you have to prove that for all $m$ and $\alpha$ that there exists an $n$ such ...
31
votes
11answers
12k views

Do factorials really grow faster than exponential functions?

Having trouble understanding this. Is there anyway to prove it?
3
votes
0answers
96 views

Prove (*) by induction on k.

Challenge: For linear systems with constant coefficients, in some sense we "never need more" than the so-called exponential polynomials, meaning expressions in the form $$\sum_{i=1}^m ...
1
vote
3answers
3k views

Find the limit of exponent/factorial sequence [duplicate]

Possible Duplicate: Prove that $\lim \limits_{n \to \infty} \frac{x^n}{n!} = 0$, $x \in \Bbb R$. Finding $\lim_{n \to \infty} \frac{\sqrt{n!}}{2^n}$ I don't know how to even stoke it... ...
3
votes
6answers
525 views

Proof by contradiction that $n!$ is not $O(2^n)$

I am having issues with this proof: Prove by contradiction that $n! \ne O(2^n)$. From what I understand, we are supposed to use a previous proof (which successfully proved that $2^n = O(n!)$) to find ...
2
votes
1answer
51 views

$t > 2n^2 \implies t!>n^t$ for $n,t \in \mathbb{N}$

I have come across this in a proof: If $t>2n^2$ then, $$t!>(n^2)^{t-n^2}=n^tn^{t-2n^2}>n^t$$ Obviously, this is much help to determine the relationship between factorials and exponential, ...
0
votes
1answer
147 views

The mathematics underlying this baroque expression of the double-factorial

On the OEIS page for the double factorial, there are three ways of getting the sequence in PARI. One of them is this curious bit of PARI code: ...