(Subscript[ϕ, 0])[x_] := (E^x);
(Subscript[ϕ, k_])[x_] := Module[{t},Simplify[(((E^t)-Normal@Series[(E^t),{t,0,k-1}])/(t^k)) /. t->x]];

Table[(Subscript[ϕ, k])[x], {k,0,3}]

ser=Series[((μ t)/ℏ)((A^(-1))(Subscript[ϕ, 1])[μ(q-1)x]-1)-(μ/ℏ)(A (Subscript[ϕ, 1])[μ((q^(-1))-1)x]-1) /. {A->(E^(-ℏ β a)), q->(E^(ℏ α β))}, {β,0,5}]

Collect[Normal@ser, β, Factor]

GS1=-μ (-1+(((E^(-a β ℏ)) (-1+(E^((-1+(E^(-α β ℏ))) x μ ℏ))))/((-1+(E^(-α β ℏ))) x μ ℏ)))+T μ (-1+(((E^(a β ℏ)) (-1+(E^((-1+(E^(α β ℏ))) x μ ℏ))))/((-1+(E^(α β ℏ))) x μ ℏ)))

GS2=Collect[Normal@Series[GS1, {β,0,3}], β, Factor]

$TD=5; Series[GS1-GS2, {ℏ,0,$TD}]

Series[(E^x)-1,{x,0,3}]

Series[x((E^x)-1),{x,0,3}]

Series[(x^7),{x,0,4}]