Re: binomial expansion of a fraction power

From: David W. Cantrell (DWCantrell_at_sigmaxi.org)
Date: 02/27/05 

Date: 27 Feb 2005 19:41:35 GMT

Virgil <ITSnetNOTcom#virgil@COMCAST.com> wrote:
> In article <20050227085722.234$ob@newsreader.com>,
> David W. Cantrell <DWCantrell@sigmaxi.org> wrote:
>
> > zyf <zhangyunfeng@gmail.com> wrote:
> > > How to do a binomial expansion to the equation below?
> > >
> > > P(x,t) r-1 v pow
> > > ------- = ( 1 + ----- * ---- )
> > > P0 2 a0
> > >
> > > where 'pow' is a fraction number.
> > >
> > > This equation is said may be expansion to third-order form, anybody
> > > know how to?
> >
> > Use equation (2) or one of the following equations at
> > <http://mathworld.wolfram.com/BinomialSeries.html>.
> >
> > BTW, I don't like the opening sentence in that MathWorld entry: "The
> > binomial series is the finite sum ..." As I recall, the term "binomial
> > series" is most often used when the expansion is _not_ finite.
>
> I can't say whether "most often" is correct, but the series is
> certainly _not_finite_ frequently enough to falsify the MathWorld entry.

Agreed (and so I'm sending a copy of this to Eric Weisstein).

> A word of caution,however!
> The expansion for '(1+x)^pow' only converges when x is less than 1 in
> absolute value, and only converges rapidly when x is close to zero, the
> closer, the faster.

Well, to be picky, it also converges when |x| = 1 if pow > 0.

David