Function To Power Series

Question

I am trying to convert this function into a power series and can't figure it out.

$f(x) = 4 e^{-5x}$

I have calculated the first four terms to be $4-20x+50x^{2}-\frac{250x^{3}}{3}$.

Answer 1

Hint: $\displaystyle e^x=1+x+\frac12x^2+\frac1{3!}x^3+\cdots$

Answer 2

By definition, $$ e^{x} = \sum_{n=1}^{\infty} \frac{x^{n}}{n!} \implies e^{-5x}= \sum_{n=1}^{\infty} \frac{(-5x)^{n}}{n!}= \sum_{n=1}^{\infty} \frac{(-1)^{n}(5x)^{n}}{n!}.$$ Since the summation is linear, multiplying $f(x)$ by $4$ results in multiplying the series representation by $4$ and finally, $$ f(x) = 4\sum_{n=1}^{\infty} \frac{(-1)^{n}(5x)^{n}}{n!}.$$ If you want the extended power series, just substitute the values of $n$ accordingly.