Wikis
Search

Binomial approximation: Wikis

Note: Many of our articles have direct quotes from sources you can cite, within the Wikipedia article! This article doesn't yet, but we're working on it! See more info or our list of citable articles.

Categories: Factorial and binomial topics

Encyclopedia

Updated live from Wikipedia, last check: May 30, 2012 13:33 UTC (41 seconds ago)

From Wikipedia, the free encyclopedia

The binomial approximation is useful for approximately calculating powers of numbers close to 1. It states that if $x$ is a real number close to 0 and $α$ is a real number, then

$(1 + x)^\alpha \approx 1 + \alpha x.$

This approximation can be obtained by using the binomial theorem and ignoring the terms beyond the first two.

The left-hand side of this relation is always greater than or equal to the right-hand side for $x > - 1$ and $α$ a non-negative integer, by Bernoulli's inequality.

Derivation using Mellin Transform

$M(p) = \int^\infty_0 (1+\alpha x)^{-\gamma}x^{p-1}dx$

Let $y=\alpha x\,$

$M(p) = \alpha^{-p}\int^\infty_0 (1+y)^{-\gamma}y^{p-1}dy$

Let y=z/(1-z)

$M(p) = \alpha^{-p}\int^1_0(1-z)^{\gamma-p-1}z^{p-1} dz$

$= \alpha^{-p}B(\gamma-p,p)\,$

$= \alpha^{-p}\frac{\Gamma(\gamma-p)\Gamma(p)}{\Gamma(\gamma)}.$

Using the inverse Mellin transform:

$(1+\alpha x)^{-\gamma}=\frac{1}{2\pi i}\int^{c+i\infty}_{c-i\infty}(x\alpha)^{-p}\frac{\Gamma(\gamma-p)\Gamma(p)}{\Gamma(\gamma)}dp$

Closing this integral to the left, which converges for $|\alpha x|<1\,$ , we get:

$(1+\alpha x)^{-\gamma}=\Sigma_{n=0}^{\infty}(\alpha x)^n \frac{(-1)^n}{n!}\frac{\Gamma(\gamma+n)}{\Gamma(\gamma)}$

$=1-\alpha x \gamma+(1/2)(\alpha x)^2 (\gamma+1)\gamma-...\,$

Categories: Factorial and binomial topics

The text of the above Wikipedia article is available under the Creative Commons Attribution-ShareAlike License. This content and its associated elements are made available under the same license where attribution must include acknowledgement of The Full Wiki as the source on the page same page with a link back to this page with no nofollow tag.