Workingexpansion =
  tau/n - (A tau^2)/(27 n^3) - (B tau^3)/(5832 n^5);
CoefficientList[Workingexpansion, tau][[2]]

1/n

Assuming[
  n ∈ Integers,
  TrueQ[Simplify[CoefficientList[Workingexpansion, tau][[2]]] != 0]]

Assuming[
 n ∈ Integers,
 TrueQ[
  Simplify[
   CoefficientList[Workingexpansion, tau][[2]] != 0
   ]
  ]
 ]