## Estimate $\frac 1 {a+b} + \frac 1 {(a+b)(a+2b)} + \cdots + \frac 1 {(a+b)\cdots(a+kb)} + \cdots$

Suppose $$a>1,b>0$$ are real numbers. Consider the summation of the infinite series: $$S=\frac 1 {a+b} + \frac 1 {(a+b)(a+2b)} + \cdots + \frac 1 {(a+b)\cdots(a+kb)} + \cdots$$ How can I give a tight estimation on the summation? Apparently, one can get the upper bound: $$S\le \sum_{k=1} ^\infty \frac 1 {(a+b)^k}=\frac 1 {a+b-1}$$ But it is not tight enough. For example, fix $$a\rightarrow 1$$, and $$b=0.001$$, then $$S=38.969939$$, it seems that $$S=O(\sqrt{1/b})$$. Another example: $$a=1$$, and $$b=0.00001$$,$$S=395.039235$$.