# Asymptotic expansion of $e^x$ other than Taylor series

Are there any asymptotic expansions of $e^x$ as $x$ goes to 0 other than Taylor series?

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## 1 Answer

The most elemantary ones are maybe $$1 + x \leq e^x \leq \frac{1}{1-x}$$ for $|x| < 1$. (The lower bound holds for all $x \in \mathbb{R}$.) These can be refined by using $e^x = e^{(x/n) \cdot n}$ into $$\left(1+\frac{x}{n} \right)^n \leq e^x \leq \left( 1 - \frac{x}{n} \right)^{-n} = \left( 1 + \frac{x}{n - x} \right)^n$$ for $n \geq 1$ and $|x| < n$.

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What you give are asymptotically same functions. What I'd like to know are expansions. If my understanding of your answer is incorrect, could you give more details? – jun.ki Dec 13 '12 at 13:39
Maybe I misinterpreted your question. – WimC Dec 13 '12 at 14:36