# Help with induction proof of a finite summation

I am stuck trying to prove the following statement:

Let $$N,J \in \mathbb{N}\cup\{0\}$$ with $$N \geq J$$ and let $$k_1,...,k_N\in \mathbb{R}$$. I define ($$i$$ is the imaginary unit) $$R_J := \prod_{m=1}^J \frac{-i}{\sum_{j=1}^{m} k_{J+1-j}}, \qquad L_{N,J} := \prod_{m=J+1}^N \frac{i}{\sum_{j=J+1}^{m} k_j}, \qquad R_0 = L_{N,N} =1.$$ Then we have
$$\sum_{J= 0}^N R_J L_{N,J} = 0 ~\text{for } N\geq 1.$$

The case $$N=1$$ is trivial. For the induction step this is how far I have got:
Suppose the statement is true for $$N-1$$. Then
$$\sum_{J= 0}^N R_J L_{N,J} = \sum_{J= 0}^{N-1} R_J L_{N,J} + R_N = \sum_{J= 0}^{N-1} R_J L_{N-1,J} \Big{(}\frac{i}{k_{J+1} + ... + k_N} + C \Big{)} + R_N,$$ where $$C$$ can be any complex number that does not depend on $$J$$. Maybe one has to choose $$C$$ in some clever way to show that the whole expression gives zero...

I checked this statement for large values of $$N$$ in mathematica, so it should be correct. I think the problem should not be very hard, anyone see the solution?