In his most recent post, Prof. Tao shared the following elementary discovery:
He first discovered this empirically in the course of his research, and then found a short proof. He didn’t share the proof on his blog yet, and instead challenged the readers to find their own proofs and observations. In this short post I’ll share a nice solution of mine. (There are some partial solutions I might add later, if I’ll be able to complete them).
The key is Leibniz’ rule. We begin by writing as a product: . We need to differentiate this product times, and we do this by applying Leibniz’ rule repeatedly:
(The notation stands for “falling factorial”: .)
The crucial observation is that:
Given this observation (which we’ll explain shortly), the even case is immediate and the odd case follows directly from the binomial theorem:
To explain the observation, note that in the even case contains the term whenever . In the odd case we need to write explicitly the terms in all 3 expressions (which are given as products), and seeing that they are the same (up to reordering and\or sign) – if we assume (w.l.o.g) we find:
There’s no clever combinatorial explanation\algebraic manipulation needed. This is why I think of this as a tautological identity, although it is surprising at first sight.
(This post was converted from tex to html using the wonderful free tool “LaTeX to WordPress”. You can find a PDF version in the “Documents” section.)
Stirling’s approximation is the following (somewhat surprising) approximation of the factorial, , using elementary functions:
( means: the ratio tends to 1 as goes to infinity.) It was proved in 1730 by the Scottish mathematician James Stirling.
This approximation has many applications, among them – estimation of binomial and multinomial coefficients. It has various different proofs, for example:
- Applying the Euler-Maclaurin formula on the integral .
- Using Cauchy’s formula from complex analysis to extract the coefficients of : .
Those proofs are not complicated at all, but they are not too elementary either. In this short note I will present, using guided exercises, 5 different proofs for weaker versions of Stirling’s approximation. In most of the real-life applications, the weaker versions are more than enough. The proofs are self-contained and completely elementary – nothing more than basic calculus (integration and differentiation, Taylor series, definitions of and ) is required. Continue reading
So here I am, starting my own maths blog. In English, in the world wide web, available for anyone interested in maths and sciene and has an internet connection.
I am a little bit excited. Continue reading