Are there any tricks to solving this sum?

Question

Suppose that I have a generic, real function $f(x)$ where $x$ is a positive integer. Suppose further that $$\sum_{i=1}^{x} f(i)=B(x),$$ $$\sum_{i=1}^{x} \Phi_i=A,$$ for some $\Phi_i>0$, where $A>0$ is constant and $B(x)$ is a positive function. My question is this: How do I solve the sum \begin{equation} \label{a} \sum_{i=1}^{x} \Phi_if(i) \end{equation} to leading error? I know that the sum itself can not be solved analytically unless all the $\Phi_i$ are known, which they aren't. So my hope is to find a solution of the form $\sum_{i=1}^{x} \Phi_if(i)=g(B(x),A)+ O(\cdots),$ where $g$ is a function and the $O$-symbol is big-$O$ notation. My guess is that $$g(B(x), A)=\frac{A}{x} B(x),$$ but I am struggling to find the correspond $O(\cdots)$. I would consult literature for this if I knew what to ask, but I am not even sure what field of mathematics this falls under. If anyone has any ideas on how to solve the problem or on literature that may lead me in the right direction, I would appreciate it.