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.

The mathematician claims to be able to give money to an infinite number of people if the first person takes $100 and the subsequent person is given half the amount.

The amount that is being given out is 100, 50, 25, 12.5, 6.25 ...

This is a geometric series with first term a = 100 and common ratio r = (1/2). The sum of an infinite terms of a geometric series with common ratio less than one is `S_oo = a/(1 - r)` . The total amount that the mathematician has to give out is only equal to `100/(1 - 1/2)` = 2*100 = 200.

**The mathematician can do what he claims if he has $200 with himself.**

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