real analysis – Simplify $f(x) + bcdot f(x+a) + b^2 cdot f(x+2a)+…$

Is it possible to simplify the following expression:
$$
S = f(x) + bcdot f(x+a) + b^2 cdot f(x+2a)+…
$$

under the assumption that $f$ is infinitely smooth and defined on compact support (in other words $f$ is non-zero only on some interval $(-d,d)$) and $a,b$ are real numbers? I am also interested in the case when $a,b$ are either very small $(to 0)$ or very large $(to infty)$. For example, if $a$ is very small number, we can expand $f$ into Taylor series and get:
begin{align}
S &approx f(x) left( 1+b+b^2 + … right) + abf'(x)left(1+2b+3b^2+…right)\
&= frac{f(x)}{1-b} + frac{abf'(x)}{(1-b)^2}+…
end{align}