An interesting series converging to a constant Let $K>0$ be a constant. Suppose $\{z_n\}_{n=1}^\infty$ is a non-decreasing positive sequence. Then the series
$$\sum_{n=1}^\infty\frac{z_n}{(K+z_1)(K+z_2)\cdots(K+z_n)}K^n=K$$ 
This is a quite interesting result as the series is convergent and the limit doesn't depend on the choice of $\{z_n\}_{n=1}^\infty$, as long as it is non-decreasing and positive
I have run computer simulations and this result seems to hold. However, I am not sure how to prove it. 

The series can be transformed into a telescoping series in the following way:
$$
\sum_{n=1}^{\infty}\frac{z_n}{\prod_{j=1}^n(K+z_j)}K^n=\sum_{n=1}^{\infty}\left(\frac{K^n}{\prod_{j=1}^{n-1}(K+z_j)}-\frac{K^{n+1}}{\prod_{j=1}^n(K+z_j)}\right)=K
$$
