Find some infinite series dependency

Question

Let $$x(a)=\sum_{n=1}^{\infty}a_n a^n$$and$$y(a)=\sum_{n=1}^{\infty}\frac{a_na^n} {4n}$$ can I find $y(x)$ and how?

Answer 1

$y=f(x)/4$ when $f$ means that in every index of the sum you divide by $n$ so actually $y=\int \frac{xda}{4a}$, that because when you divide by $a$ you decrease every power of $a$ by one then, when you apply the integral you increase those powers back and divide by the power.