In my studies, I stumbled across the following: $$p(t) = f(t) + f(t) * g(t) + f(t) * g(t) * g(t) + f(t) * g(t) * g(t) * g(t) ...$$ where the Laplace transform of f(t) and g(t) is known, * is the convolution of the listed functions: $$ \int_0^t{f(t)g(t-s)ds} $$
Can I also write the function p after Laplace transform like this? $$ P(s) = F(s) + F(s)G(s) + F(s)G(s)G(s) + F(s)G(s)G(s)G(s)+ \cdots$$ Hence: $$ P(s) = F(s)G(s)^0 + F(s)G(s)^1 + F(s)G(s)^2 + F(s)G(s)^3+ \cdots$$ So: $$ P(s) = F(s)\sum_{n=0}^{\infty}{G(s)^n} $$
Or is this an unqualified simplification? Thanks