dividing a unit in several different ways There is a king who wants to divide his kingdom between his infinity of daughters. Suppose he wants to divide the kingdom evenly. It seems that under such conditions, each of the daughters gets an infinitely small piece of the kingdom. Suppose on the other hand that the king is a favouritist. He gives half of the kingdom to first daughter half of the rest to the second daughter and so on. In this case each daughter gets a finite non-zero part of the kingdom. So in the second case each daughter is getting more and yet we are dividing the same quantity into the same number of pieces. How is this possible?

It is just not possible to divide the kingdom evenly between infinitely many daughters. There does not exist a convergent series $\Sigma_{n=1}^{\infty} a_n$ with all $a_n$ being equal. The second case is possible because the series $\Sigma_{n=1}^{\infty} 2^{-n}$ is convergent.