Converting power series to an equation

Question

A power series is given by:

$$\sum_{n=0}^{\infty} \frac{(1+5^n)(x^n)}{n!}$$

Write down the series as a function of $x$ with the help of $e$ .

I've tried substituting $n = 0,1,2,3... $ and it seems like a Taylor series but it doesn't seem like that's the way to do it. I'm pretty clueless at the moment as I haven't done anything like this yet, any suggestions?

Answer 1

$(1+5^n)(x^n)=x^n+5^nx^n=x^n+(5x)^n$

$$\implies\sum_{n=0}^{\infty} \frac{(1+5^n)(x^n)}{n!} =\sum_{n=0}^{\infty} \frac{x^n}{n!}+\sum_{n=0}^{\infty} \frac{(5x)^n}{n!}$$

Now $\sum_{r=0}^\infty\dfrac{y^r}{r!}=e^y$