# 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}