Maple produces a closed-form expression for the sum under consideration by

sum(1/product(b*j+a, j = 1 .. k), k = 0 .. infinity))/(a+b)  assuming a>1,b>0;

$${\frac {a{{\rm e}^{{b}^{-1}}}}{a+b}{b}^{{\frac {a-b}{b}}} \left( - \Gamma \left( {\frac {a}{b}},{b}^{-1} \right) +\Gamma \left( {\frac {a }{b}} \right) \right) } $$

Maple produces a closed-form expression for the sum under consideration by

sum(1/product(b*j+a, j = 1 .. k), k = 0 .. infinity))/(a+b)  assuming a>1,b>0;

$${\frac {a{{\rm e}^{{b}^{-1}}}}{a+b}{b}^{{\frac {a-b}{b}}} \left( - \Gamma \left( {\frac {a}{b}},{b}^{-1} \right) +\Gamma \left( {\frac {a }{b}} \right) \right) } $$

Addition. So does Mathematica through

Sum[1/Product[b*j + a, {j, 1, k}],{k, 0, Infinity},Assumptions -> b > 0 && a > 1]/(a + b)

Maple produces a closed-form expression for the sum under consideration by

sum(1/(product((a+b)*(b*j+a), j = 0 .. k)), k = 0 .. infinity) assuming a>1,b>0;

$${\frac {1}{ \left( a+b \right) b}{{\rm e}^{{\frac {1}{ \left( a+b \right) b}}}} \left( \Gamma \left( {\frac {a}{b}} \right) -\Gamma \left( {\frac {a}{b}},{\frac {1}{ \left( a+b \right) b}} \right) \right) \left( \left( {\frac {1}{ \left( a+b \right) b}} \right) ^{ {\frac {a}{b}}} \right) ^{-1}} $$

Maple produces a closed-form expression for the sum under consideration by

sum(1/product(b*j+a, j = 1 .. k), k = 0 .. infinity))/(a+b)  assuming a>1,b>0;

$${\frac {a{{\rm e}^{{b}^{-1}}}}{a+b}{b}^{{\frac {a-b}{b}}} \left( - \Gamma \left( {\frac {a}{b}},{b}^{-1} \right) +\Gamma \left( {\frac {a }{b}} \right) \right) } $$