# Tagged Questions

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)} = ...
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 ...
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 ...
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 $t>2$, ...
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 ...
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 ...
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 ...
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 $\log {m \choose n}$?
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 ...
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 ...
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