I am very desperately longing to know if there is a explicit relationship between $$F(n)=f(1)+f(2)+...+f(n)$$ and $$G(x)=\sum_{k=1}^{\infty}f(k)x^k$$ Assuming we can let $f$ be a sufficiently well behaved function, surely there must be a way of linking the growth rate of the $F$ and $G$. My guess was
$$F(n)=f(1)+...+f(n)\to \sum_{k=1}^{\infty}f(k)(1-\frac{1}{n})^k=G(1-\frac{1}{n})$$
Which is roughly correct for some cases of $f$ I tried. (Roughly in the sense that the ratio converged). If a link can be found, it should be particularly useful as it would mean that knowing the growth rate of a ordinary generating function will tell us information about its coefficients (which can be arithmetical functions like $\pi(n)$ etc...).