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Find the limit of sequence

  1. Jun 14, 2012 #1
    1. The problem statement, all variables and given/known data
    I want to find the limit of ƩK(n+m,n)zn
    K(a,b) being the binomial coefficient.

    2. Relevant equations
    Cauchy root test?


    3. The attempt at a solution

    Trying the cauchy root test I get:

    1/R = limn->∞[(K(n+m,n))½]

    But what do I do from here?
     
  2. jcsd
  3. Jun 14, 2012 #2

    HallsofIvy

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    The "Cauchy root test" tells you whether or not a series converges. It says nothing about what it converges to. If I read this correctly, you have
    [tex]\sum \begin{pmatrix}n+m \\ n\end{pmatrix}z^n[/tex]

    The sum is over n with m fixed? And it is a finite sum? n goes from 0 to what?
     
  4. Jun 14, 2012 #3
    well maybe I named it wrong, but I meant the formula stated above, which gives an explicit expression for the radius of convergence, R.
    And the sum is from zero to infinity. Sorry for the lack of information :)
     
  5. Jun 14, 2012 #4

    Ray Vickson

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    If you set z = -t, the coefficient of t^n is the "negative binomial" coefficient:
    [tex] (-1)^n {n+m \choose n} = {-m \choose n}.[/tex] That should allow you to evaluate the sum explicitly.

    RGV
     
    Last edited: Jun 14, 2012
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