How do I convert a function to sigma notation when there is a nested and lagging summation within the function?

Question

I am trying to simplify the following function into summation form: $$f(x) = \frac{x_1}{x_0}+\frac{x_2}{x_0-x_1}+\frac{x_3}{x_0-x_1-x_2}+...+\frac{x_n}{x_0-x_1-x_2-...-x_{n-1}}.$$

However, I am unsure as to how to best simplify the notation as it involves two parts that are iterating, the denominator summation lagging behind the numerator's summation. The closest that I can think of is

$$f(x)=\sum^{n}_{i=1}\frac{x_i}{x_0-\sum_{j=0}^{i-1}x_j}$$

but I think this simplification lacks clarity and my gut (and brain) is saying that this isn't correct use of notation either.

If necessary for simplification, here is more context and assumptions. This equation was created through conservation of mass, thus the following conditions must hold:

1. All values of $x$ are positive
2. The initial state, $x_0$, cannot be zero
3. $x_0\geq \sum^{n}_{k=1} x_k $

Thanks in advance for any advice or help <3