Here's a taylor series problem I've been working on. I'll list a few steps to the problem and tell you guys where I'm getting stuck. Thanks in advance for the help.
So my questions builds off the fact that
$ e^x = \sum_{n=0}^{\infty}\frac{x^n}{n!}$
and we are asked to find the taylor series of the following function:
$f(x) = (2x-3)\cdot e^{5x}$ around a = 0
So I first decided to calculate the taylor series for $e^{5x}$ by generating a few terms and noticing the pattern. I then found the following series to represent $e^{5x}$
$ e^{5x} = \sum_{n=0}^{\infty}\frac{5^n}{n!} \cdot x^n$
Next I know I must multiply this series by (2x-3) somehow so I begin like this:
$(2x-3) \cdot \sum_{n=0}^{\infty}\frac{5^n}{n!} \cdot x^n$
$\sum_{n=0}^{\infty}\frac{(2x-3)5^n}{n!} \cdot x^n$
My problem with this answer is that it's not in the correct form for a taylor series and must be in the form:
$\sum b_n \cdot x^n$
Does anyone know the type of manipulations I must do to convert my result to the correct form?