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Binomial series

  1. Jun 15, 2005 #1
    hello all

    I thought this might be an interesting question to ask, consider the following series
    [tex]\sum_{n=0}^{\infty}\left(\begin{array}{cc}\alpha\\n \end{array}\right)x^{n}=(1+x)^{\alpha}[/tex]
    this is known as the binomial series, whats confusing me is that how could this series exist when [tex]\alpha< n[/tex] especially when its a series that adds infinitely
    number of terms, from my understanding this [tex]\left(\begin{array}{cc}\alpha\\n \end{array}\right)[/tex] can only be evaluated when [tex]\alpha>n[/tex] please help

    thanxs
     
    Last edited: Jun 15, 2005
  2. jcsd
  3. Jun 15, 2005 #2

    matt grime

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    they are identically zero for n>alpha (assuming alpha is an integer)
     
  4. Jun 15, 2005 #3
    hello Matt
    well im still confused, what im not understanding is that if

    [tex]\left(\begin{array}{cc}\alpha\\n \end{array}\right)=\frac{\alpha !}{n!(\alpha-n)!}[/tex] and [tex]\alpha< n[/tex] then [tex]\alpha-n[/tex] is negetive how could you find the factorial of a negetive

    number, and if they were identically equal to zero i couldnt see how this
    [tex]\frac{\alpha !}{n!(\alpha-n)!}=0[/tex]
     
  5. Jun 16, 2005 #4

    matt grime

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    They are *DEFINED* to be zero. Who said that that formula holds for the cases you have a problem with?

    there is another way to define the binomial coefficient for non-integer alpha too, it's


    a(a-1)(a-2)..(a-n+1)/n!

    that is defined for all a and all integer n.
     
  6. Jun 16, 2005 #5
    yeah I get what you mean, its just that I couldnt find any site that tells what happens in that sanario so i got really confused anyway

    Thanxs Matt
     
    Last edited: Jun 16, 2005
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