1) Are you SURE that's the right construction? Why would a finite dollar bill need the product of five infinite series?
2) You just multiplied something wrong. Where would the x^50 go?
I have to figure out all of the ways to make change for a dollar. I know that it is equal to the product of the geometric series for each of the coins, or (1/(1-x))(1/(1-x^5))(1/(1-x^10))(1/(1-x^25))(1/(1-x^50)), but I am stuck there. When I multiplied it out, the highest term I got was x^19, so I don't know what to do. Is there an easier way to do this problem or a way to actually find the answer?