I consider only one Sum / Product..
Check out the following for differnt M. note the value of sum1 - sum2
M = 5;
sum1 = Collect[Expand[(\!\(
\*UnderoverscriptBox[\(\[Product]\), \(m = 0\), \(M\)]\((1 -
\*FractionBox[
SubscriptBox[\(c\), \(m\)], \(z\)])\)\)) 1], z]
tt = Table[Subscript[c, j], {j, 0, M}]
ss = Subsets[tt, {2, M + 1}]
sum2 = 1 - (\!\(
\*UnderoverscriptBox[\(\[Sum]\), \(j = 0\), \(M\)]
\*SubscriptBox[\(c\), \(j\)]\))/z + \!\(
\*UnderoverscriptBox[\(\[Sum]\), \(j = 2\), \(M + 1\)]\(
FractionBox[\(Plus @@ \((Apply[Times,
Select[ss, Length[#] == j &], {1}])\)\),
SuperscriptBox[\(z\), \(j\)]]
\*SuperscriptBox[\((\(-1\))\), \(j\)]\)\)
Simplify[sum1 - sum2]