Error terms in Taylor Series

  1. It is known that
    [tex] \sum\limits_{k = 0}^\infty {\frac{{N^k }}{{k!}}} = e^N[/tex]

    I am looking for any asymptotic approximation which gives

    [tex] \sum\limits_{k = 0}^M {\frac{{N^k }}{{k!}}} = ? [/tex]
    where [tex]M\leq N[/tex] an integer.

    This is not an homework
  2. jcsd
  3. I'm only being a little facetious if I point out that the sum is asymptotically equal to e^N.

    Want more accuracy? It's better approximated by e^N-[x^(N+1)]/(N+1)!
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