Tagged Questions
3
votes
0answers
37 views
Bound on Permutations [duplicate]
I am trying to prove the following inequality,
$$n^{(l)} = n(n-1)\cdots(n-l+1) \geq \frac{n^l}{e}\quad\text{ for }\quad 2 \leq l \leq \sqrt{n}\;.$$ So my approach is to observe that $n^{(l)} = ...
1
vote
0answers
40 views
Approximation of factorial - Stirling formula [duplicate]
Possible Duplicate:
Elementary central binomial coefficient estimates
How can I prove that
$$
\binom{n}{n/2} = \Theta\left(\frac{2^n}{\sqrt n}\right)
$$
I tried with Stirlings ...
2
votes
2answers
117 views
How many bits are in factorial?
I am interested in good integer approximation from below and from above for binary Log(N!). The question and the question provides only a general idea but not exact values.
In other words I need ...
2
votes
0answers
123 views
Generating all positive integers from three operations
This question arose on sci.math, but since almost all the competent mathematicians there have migrated here, I thought I'd give the question a wider audience. Starting with an integer $t>2$, ...
3
votes
2answers
185 views
How to approximate $\sum_{k=1}^n k!$ using Stirling's formula?
How to find summation of the first $n$ factorials,
$$1! + 2! + \cdots + n!$$
I know there's no direct formula, but how can it be estimated using Stirling's formula?
Another question :
Why can't ...
3
votes
3answers
651 views
Approximating log of factorial
I'm wondering if people had a recommendation for approximating $\log(n!)$. I've been using Stirlings formula,
$ (n + \frac{1}{2})\log(n) - n + \frac{1}{2}\log(2\pi) $
but it is not so great for ...
20
votes
9answers
2k views
What is the purpose of Stirling's approximation to a factorial?
Stirling approximation to a factorial is
$$
n! \sim \sqrt{2 \pi n} \left(\frac{n}{e}\right)^n.
$$
I wonder what benefit can be got from it?
From computational perspective (I admit I don't ...
8
votes
2answers
2k views
Approximating the logarithm of the binomial coefficient
We know that by using Stirling approximation:
$\log n! \approx n \log n$
So how to approximate $\log {m \choose n}$?
16
votes
1answer
356 views
A series problem by Knuth
I came across the following problem, known as Knuth's Series which originally was an American Mathematical Monthly problem.
Prove that $$\sum_{n=1}^\infty ...
16
votes
5answers
688 views
How best to explain the $\sqrt{2\pi n}$ term in Stirling's?
I recently showed my Algorithms class how to bound $\ln n! = \sum \ln n$ by integrals, thereby obtaining the simple factorial approximation
$$ e \left(\frac{n}{e}\right)^{n} \leq n! \leq ...
