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The limit of a series

  1. Feb 19, 2013 #1
    1. The problem statement, all variables and given/known data

    prove that lim n→∞ of [itex]\sum[/itex][itex]^{n}_{k=0}[/itex] e[itex]^{-n}[/itex] n[itex]^{k}[/itex] / k! = 1/2

    3. The attempt at a solution

    I seem to be mishandling the series. After taking n→∞, the sum of (n^k)/k! is just the taylor series expansion of e^n. Then I should get e^(-n)*e^n = 1.

    Where am I going wrong??
     
    Last edited: Feb 19, 2013
  2. jcsd
  3. Feb 19, 2013 #2

    micromass

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    You're messing up your variables.

    You say "After taking the limit [itex]n\rightarrow +\infty[/itex]". But you seem to interpret that only has "making the sum infinite".

    The expressions [itex]e^{-n}[/itex] and [itex]n^k[/itex] also depend on n. So if you take the limit as [itex]n\rightarrow +\infty[/itex], then those things don't stay the same.

    Allow me to totally butcher mathematics for a moment, but it's to make things clear. If you take the limit [itex]n\rightarrow +\infty[/itex] of your expression then you don't end up with

    [tex]e^{-n}\sum_{k=0}^{+\infty}\frac{n^k}{k!}[/tex]

    Rather, you would end up with (please forgive me)

    [tex]e^{-\infty}\sum_{k=0}^{+\infty}\frac{\infty^k}{k!}[/tex]

    The above of course makes no sense. But I think it makes the situation clear.
     
  4. Feb 19, 2013 #3

    Zondrina

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    Prove that what? That the series converges?
     
  5. Feb 19, 2013 #4
    Sorry, prove the limit = 1/2 (could have sworn I had written it).

    Micromass, thanks. I would be lying if I said I didn't suspect that was the problem, it still kind of makes me uneasy.

    Anyways now I am trying to replace the sum by an integral (Euleur-Maclaurin), and I get:

    [itex]\sum[/itex][itex]^{n}_{k=0}[/itex] e[itex]^{-n}[/itex][itex]\frac{n^k}{k!}[/itex] = 1/2 + [itex]\frac{e^{-n}}{2}[/itex][itex]\frac{n^n}{(n!)}[/itex] + [itex]\int[/itex][itex]^{n}_{0}[/itex] e[itex]^{-n}[/itex][itex]\frac{n^k}{k!}[/itex]dk

    so I have hopes because the 1/2 is there to stay, the 2nd term probably goes to zero after taking the limit, same with the integral except I don't know how to handle the factorial in the integral. Now I am reading on Gamma functions.. maybe that will help.
     
    Last edited: Feb 19, 2013
  6. Feb 19, 2013 #5

    micromass

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    It may sound like a strange question, but do you know the Central Limit Theorem and Poisson distributions?
     
  7. Feb 19, 2013 #6
    No, I haven't learned them.. although googling Poission distributions.. it seems it is related to my problem.

    In applying for a Masters in Fluid Dynamics (Math) after getting a Physics Undergrad, one of my potential supervisors gave me a set of problems to 'check me out'. I killed most of them without too much sweat and tears, but this one really I'm having a hard time with.

    Anyways,I will read about Poisson distributions and Central Limit Theorem.
     
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