Approximating a discrete function with continuous function.

Question

$$\sum^n_{i=1}\frac{x^i}{(n-i)!}$$ I tried approximating it but i am not able to find any series similar to this . I need to find a function which can be used in place of this series. Thanks!

Answer 1

$\sum^n_{i=0}\frac{x^i}{(n-i)!}=\sum^n_{i=0}\frac{x^{n-i}}{i!}=x^{n} \sum^n_{i=0}\frac{x^{-i}}{i!}$. You can approximate by $x^{n}e^{1/x}$ provided $x$ is not close to $0$.

Answer 2

$$S=\sum^n_{i=0}\frac{x^i}{(n-i)!}=x^{n} \sum^n_{j=0}\frac{x^{-j}}{j!}\quad;\quad i=n-j$$ $$S=\frac{x^n e^{1/x}}{n!}\Gamma\left(n+1\:,\:\frac{1}{x}\right)$$ $\Gamma(*,*)$ is the Incomplete Gamma function. This is a direct consequence of the series definition of the Incomplete Gamma function. http://mathworld.wolfram.com/IncompleteGammaFunction.html

They are a lot of approximates depending the range of the variable. Your question is not clear enough. Approximate in what domain ? Asymptotic approximate ? This is a too wide question.

You can use the first terms of a lot of series. For example http://functions.wolfram.com/GammaBetaErf/Gamma2/06/