Trying to understand an expansion/limit from geometric sum to exponentials, what kind of rule is at play?

Question

Can someone help me understand what's going on here?

This is for a problem involving moment generating functions, which is related to statistics and probability, but I figured it was more of a math questions. The whole expansion is below:

Answer 1

Rewrite the right-hand side using the Taylor expansion for $\exp$.

Answer 2

Hint

$$A=1+\sum_{n=1}^\infty \frac{t^n}{n!}\big(\frac 19+3^{n-1}\big)=1+\frac 19\sum_{n=1}^\infty \frac{t^n}{n!}+\frac 13\sum_{n=1}^\infty \frac{(3t)^n}{n!}$$ $$A=1+\frac 19 \Big(\sum_{n=0}^\infty \frac{t^n}{n!}-1\Big)+\frac 13\Big(\sum_{n=0}^\infty \frac{(3t)^n}{n!}-1\Big)$$ Now, remember that $$e^x=\sum_{n=0}^\infty \frac{x^n}{n!}$$