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The power series representation of $e^x = \sum \limits_{k=0}^{\infty} \frac{x^k}{k!}$. Can I use this approximation for $e^{-x} = 1/e^x = 1/\sum \limits_{k=0}^{\infty} \frac{x^k}{k!}$ instead of $e^{-x} = \sum \limits_{k=0}^{\infty} \frac{(-1)^k x^k}{k!}$. What is the difference between two approximation?

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    I have not done any thinking about it, but in in some cases at least the $1/(1+x+\cdots)$ approach seems to give a better approximation for given number of terms. It would be interesting to know something general. Nice question!2011-11-13

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The two are equivalent. If you know about power series multiplication, you can see that

$ \left(\sum_{k=0}^\infty \frac{x^k}{k!}\right)\cdot \left(\sum_{k=0}^\infty \frac{(-1)^kx^k}{k!}\right)=1. $

If you expand the product of the two power series, terms will cancel, leaving you with $1$.