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