Is this sum integer? $b! \pi+1/(b+1)+1/(b+2)(b+1)+...$, where $b \neq 0$. Is $S$ an integer ?  
$S= \: b! \:\pi+\frac{1}{b+1}+\frac{1}{(b+2)(b+1)} + \frac{1}{(b+3)(b+2)(b+1)} + ...$
$b \neq 0$. 
Also, from here Is this sum rational or not? $1/(q+1)+1/(q+2)(q+1)...$ where $q$ is an integer
$\frac{1}{b+1}+\frac{1}{(b+2)(b+1)} + \frac{1}{(b+3)(b+2)(b+1)} + ...$ is irrational and between $(0,1)$.

Hint:
$$S_b=\frac{1}{b+1}+\frac{1}{(b+2)(b+1)} + \frac{1}{(b+3)(b+2)(b+1)} + ...+\: b! \:\pi=\dfrac{1}{b+1}(1+\frac{1}{b+2}+\frac{1}{(b+3)(b+2)} + \frac{1}{(b+4)(b+3)(b+2)} + ...+\: (b+1)! \:\pi)=\dfrac{S_{b+1}+1}{b+1}\to\\S_b=\dfrac{S_{b+1}+1}{b+1}$$can you finish now?
