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Exponential Identities

  1. Mar 31, 2014 #1
    Hi All,

    I am struggling to prove the following identity

    $$ 1 + y + \frac{1}{2!}y^2 + \frac{1}{3!}y^3 + \dots = lim_{N \to \infty} \sum_{r=0}^{N} \frac {N!}{r! (N-r)!} \frac{y}{N}^{r} = lim_{N \to \infty} (1 + \frac{y}{N})^N $$



    any hint would the most appreciated. I understand the left-most term is the Taylor series for the exponential function, and the right-most term is also used as a definition of such function, yet I would like to know how the two are explicitly shown to be equivalent.

    Thanks as usual
     
  2. jcsd
  3. Mar 31, 2014 #2

    Erland

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    There is a typo in the middle formula. It should be: ## \lim_{N \to \infty} \sum_{r=0}^{N} \frac {N!}{r! (N-r)!} (\frac{y}{N})^{r}##.

    It is then easy to see that this is the binomial expansion of the right side.
    One also sees that for each fixed ##r##, the ##r##-term in the middle sum tends to ##y^r/r!## as ##N \to\infty##.

    It remains to convince oneself that everything works out with the limits.
     
    Last edited: Mar 31, 2014
  4. Mar 31, 2014 #3
    Erland,
    many thanks for pointing this out, I am a little bit closer now to understanding, many thanks
    !
     
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