Binominal series

by Nikitin
Tags: binominal, series
Nikitin is offline
Dec14-12, 02:05 PM
P: 588
Hello. I'm revising the material in preparation for the exam, and I found something I fail at understanding.

When defining binomial series,, why is the sum of the binomial "(m k)" going from 1 to ? Shouldn't it instead be going from 1 to m (the function can only be differentiated m times)?

Afterall, binomial series are a form of taylor series, and a taylor series of a function can't have infinite terms when the function can only be differentiated a finite amount of times.
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Michael Redei
Michael Redei is offline
Dec14-12, 02:25 PM
P: 181
Which function can only be differentiated m times? The binomial series is the Taylor series at x=0 of (1+x)α for some complex number α. And (1+x)α can be differentiated infinitely often, unless α is a non-negative integer.
jbriggs444 is offline
Dec14-12, 02:26 PM
P: 748
Just because all but finitely many terms are zero does not mean that you cannot consider a series as having infinitely many terms. The function f(x) = (1+x)^m can be differentiated more than m times. It's just that all of the derivitives are eventually zero.

It looks to me like a choice to make the summation look more like the generic form of the Taylor series (which it is, after all) rather than an equally accurate truncation thereof.

Nikitin is offline
Dec14-12, 04:03 PM
P: 588

Binominal series

Michael: oops, sorry. My book used the notation of m instead of a, and I have little knowledge of complex numbers (I'm only doing my 1st semester).

jbriggs: yeh, so it's just a formal thing? Allright, that's good enough 4 me :)
awkward is offline
Dec14-12, 06:19 PM
P: 325
The binomial series for [itex](1+x)^\alpha[/itex] has infinitely many terms except when [itex]\alpha[/itex] is a non-negative integer (even when [itex]\alpha[/itex] is real).

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