The mathematian would be able to give out money to infinite number of people, as described above, if the unit of money was infinitely divisible. Then, in fact, he could give out $200 because it is a sum of a geometric sequence.
In real world, however, once he gets to 1/2 of $6.25, he would have to give out $3.125, which is 3 dollars and 12 and a half cents. Further division would result in a quarter of a cent, and so on. Such units of money do not exist.
Thus, while it is mathematically possible to write 200 as a sum of infinite series, it is physically impossible to break down 20000 descrete units (cents) into infinite number of groups because one cannot produce half or other fraction of a cent.