lambda1=(((5-Sqrt[33]))/2);
lambda2=(((5+Sqrt[33]))/2);
a1[t_]:=-Series[(((lambda2 (E^(lambda1 t))-lambda1 (E^(lambda2 t))))/(lambda1-lambda2)),{t,0,6}] (* note negative sign *)
a2[t_]:=Series[((((E^(lambda1 t))-(E^(lambda2 t))))/(lambda1-lambda2)),{t,0,6}]
a1[t]
a2[t]