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Convergence of the following series as N goes to infty

  1. Nov 25, 2008 #1
    hi there!

    I want to show the convergence of the following series as N goes to infty.

    [tex]\displaystyle{\sum_{k=0}^N}\frac{x^k}{k!}-\frac{n!x_n^k}{k!(n-k)!n^k}[/tex],

    x_n is a sequence such that. lim(n->oo)x_n = x, but I´m interested in big N

    I ´m not allowed to use the limit definition of exp(x)

    What I want to do (but am not sure if it´s correct) is to separate the sum before taking the limit N->oo and after that take it, so that the first term converges to exp(x) an the convergence of the second term I can show with the ratio test
     
  2. jcsd
  3. Nov 25, 2008 #2

    lurflurf

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    Re: series

    Are you sure N and n are not related?
    What is your convention when k>N?
    The usual convention is that nCk=0
     
  4. Nov 25, 2008 #3
    Re: series

    I don't know if this will help, but did you notice that the [tex]\frac{n!}{k!(n-k)!}[/tex] is the binomial coefficient? Might be worth something...
     
  5. Nov 25, 2008 #4
    Re: series

    lurflurf, sorry I forgot to say that n>=N, but it really comes up to N not n.

    JG89, thanks, it is also given as the binomial coefficient, I wrote it that way cuz I don´t know how to write it in latex language :(


    could you say if I can do it the way I described it, or it´s somehow wrong to do like that?
     
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