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 a non-decreasing and positive sequence.


I have run computer simulations and this result seems to hold. However, I am not sure how to prove it. 



An interesting series converging to a constant
The point is that the partial sum


$$ \sum\_{n=1}^N \frac{z\_n}{(K+z\_1)\ldots(K+z\_n)} K^n = K - \frac{K^{N+1}}{(K+z\_1)\ldots(K+z\_N)} $$


as is easy to prove by induction.


