Normal[Series[
    (E^(-(λ/μ)) (λ/μ)^k Gamma[1 + k])/(k! Gamma[1 + k, λ/]) /. λ -> 1/x, {x, 0, 1},
    Assumptions -> {x > 0, μ > 0, k > 0}
  ]] /.x -> 1/λ
(* (λ/μ)^k (μ/λ)^k (Gamma[1 + k]/k! - (k μ Gamma[1 + k])/(λ k!)) *)