Error terms in Taylor Series

Nurdan

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

christianjb

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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Nurdan	Error terms in Taylor Series Tweet 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

christianjb	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)!