Sum
does not automatically collect common coefficients, but you can do it with a replacement rule:
collectCoefficientInSum :=
Sum[(coeff_)*x_, {y_, z__}] /; FreeQ[coeff, y] :>
coeff*Sum[x, {y, z}];
(Sum[Subscript[b, m] z^(-m), {m, 0, M}]/
Sum[Subscript[a, n] z^(-n), {n, 0, N}] == (Subscript[b, 0]/
Subscript[a,
0]) (Sum[(Subscript[b, m]/Subscript[b, 0]) z^(-m), {m, 0, M}]/
Sum[(Subscript[a, n]/Subscript[a, 0]) z^(-n), {n, 0,
N}])) /.
collectCoefficientInSum