Q.The co-efficient of $x^5$ in the series $e^{-2x}$ is:

Exponential Series

The exponential function $e^x$ has a power-series expansion valid for every real (and complex) number $x$:

Each term is $\frac{x^n}{n!}$, so the coefficient of $x^n$ in the expansion of $e^x$ is $\frac{1}{n!}$. Setting $x = 1$ recovers the definition of the constant $e = 1 + 1 + \frac12 + \frac16 + \cdots \approx 2.718$. Replacing $x$ by $ax$ gives the closely related series $e^{ax} = \sum_{n=0}^{\infty} \frac{a^n x^n}{n!}$, so the coefficient of $x^n$ there is $\frac{a^n}{n!}$. …