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Binominal series

by Nikitin
Tags: binominal, series
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Dec14-12, 02:05 PM
P: 632
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
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.
Dec14-12, 02:26 PM
P: 999
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.

Dec14-12, 04:03 PM
P: 632
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 :)
Dec14-12, 06:19 PM
P: 327
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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