Hello Luther,
I am not sure what you are after here. Does this not do what you wish it to?
Clear[a, b];
originalSeries =
Series[1/(1 + a)^3, {a, 0, 5}]*Series[1/Sqrt[1 + b], {b, 0, 5}]
originalSeries /. {a -> (z1 - z2)}
I imagine that you might wish to have this instead:
originalSeriesNormal = Normal[originalSeries]
originalSeriesNormal /. {a -> (z1 - z2)}
But, this may be even better for your purposes:
expansion[ax_, bx_] := Normal[Series[1/(1 + ax)^3, {ax, 0, 5}]* Series[1/Sqrt[1 + bx], {bx, 0, 5}]]
expansion[z1 - z2, b]
Integrate[expansion[z1 - z2, b], z1]
I hope this is helpful to you.
Craig