Say we have $$a\_n=\frac{n-1+b}{n-b}a\_{n-1}$$ with $a\_0>0$ and $-1<b<0$. Then it holds that $$\sum\_{n=1}^{\infty}a\_n=-\frac12 a\_0.$$ 
I could establish this using simple manupulations with $\Gamma$ and $Beta$ funtions. I was however wondering if one could do it differently. I appreciate if you share your fun on this problem with me :-)