A(x) is the generating function of the series $\{a_n\}^\infty $ ( n from 0), what generates,$f_n=(-1)^na_n$

Question

A(x) is the generating function of the series $\{a_n\}^\infty $ ( n from 0)

I am given the series $f_n=(-1)^na_n$ and need to find the function F(x).

I thought that because the function $1/(1+x)$ generates the series $\lambda n\in N.(-1)^n $ so $F(x)= A(x)/(1+x)$ , but I not sure if it generates it , what do you think ?

Answer 1

Note that $$A(-x) = \sum_{n=0}^{\infty} a_n (-x)^n = \sum_{n=0}^{\infty} (-1)^na_n x^n,$$ and so the function $F(x)$ is simply $A(-x)$.