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$ .