Finding maxima of a multivariable function

Question

I would like to find the maxima of this function: $$\frac{a_1x}{a_1+KV}+\frac{a_2x}{a_2+KV}+\frac{a_1a_2x}{(a_1+KV)(a_2+KV)}+\frac{a_3x}{a_3+KV}+\frac{a_2a_3x}{(a_2+KV)(a_3+KV)}+\frac{a_1a_2a_3x}{(a_1+KV)(a_2+KV)(a_3+KV)}+...+\frac{a_nx}{a_n+KV}+\frac{a_1a_2a_3...a_nx}{(a_1+KV)(a_2+KV)(a_3+KV)...(a_n+KV)}$$ Given $a_1+a_2+...+a_n=V'$ where $x, K, V, V'$ are constants.
For convenience, I've converted the series into a simpler form $$x\sum_{i=1}^n\sum_{j=1}^n\prod_{p=i}^j\frac{a_p}{a_p+KV}$$ I can't think of any inequality that can suit this series at the moment. If the answer is trivial enough, I would like anyone to provide tips rather than the complete answer.