Series[((1-(E^(-ϵ)))/ϵ),{ϵ,0,3}]

α=(1-(E^ϵ));
β=(1-(E^(-ϵ)));
Expand[α+β-α β]

γ[1]=1-(E^ϵ);
γ[n_] /; n>0 := Expand[γ[1]+γ[n-1]-γ[1]γ[n-1]]

γ[10]