s[r_, \[Alpha]_] := \[Alpha]*r
G[r_] := Exp[-r] (* I assume *)
f[r_] := Sin[r] (* I assume *)
xxSum[r_, \[Alpha]_] := Sum[(-s[r, \[Alpha]])^k/k!*D[G[r], {r, k}], {k, 0, 4}];
xxLAB[r_, \[Alpha]_] := xxSum[r, \[Alpha]]*f[r];
A[\[Alpha]_] := NIntegrate[xxLAB[r, \[Alpha]], {r, 100, Infinity}];
A[10^10] (* I assume alpha = 10^10 *)
(* 303.243 *)