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Binominal series

  1. Dec 14, 2012 #1
    Hello. I'm revising the material in preparation for the exam, and I found something I fail at understanding.

    When defining binomial series, http://en.wikipedia.org/wiki/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.
  2. jcsd
  3. Dec 14, 2012 #2
    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.
  4. Dec 14, 2012 #3


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    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.
  5. Dec 14, 2012 #4
    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 :)
  6. Dec 14, 2012 #5
    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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