Convergence of the infinite series

On this page we'll give some hints for proving that the infinite series mentioned in the main article converge to the values claimed (using slightly more advanced maths).

We start with the series $$S = (2 \times 1/2) + (3 \times 1/4) + (4 \times 1/8) + ... + (n \times 1/2^{(n-1)}) +...$$ First note that the series $$1/2+ 1/4+ 1/8+ ... + 1/2^n + ... $ converges to 1 (that's a standard result).

Using the ratio test you can prove that the series

$$S = (2 \times 1/2) + (3 \times 1/4) + (4 \times 1/8) + ... + (n \times 1/2^{(n-1)}) +...$$

converges, in fact it converges absolutely. We're allowed to rearrange the terms of an absolutely convergent series, as this won't change its limit.

We'll rearrange the terms of $S$ in a way that splits it up into infinitely many individual series: \begin{eqnarray*} S & = & 2/2 & + & 3/4 & + & 4/8 & + & 5/16 & ... & + & n/2^{n-1} & + & ... \\ & = & 1/2 & + & 1/4 & + & 1/8 & + & 1/16 & ... & + & 1/2^{n-1} & + & ... \\ & + & 1/2 & + & 1/4 & + & 1/8 & + & 1/16 & ... & + & 1/2^{n-1} & + & ... \\ & + & & & 1/4 & + & 1/8 & + & 1/16 & ... & + & 1/2^{n-1} & + & ... \\ & + & & & & + & 1/8 & + & 1/16 & ... & + & 1/2^{n-1} & + & ... \\ & + & & & & + & & + & 1/16 & ... & + & 1/2^{n-1} & + & ... \\ & + & & & & + & & + & & ... & + & ... & + & ... \\ \end{eqnarray*} Each of these infinitely many series is of the form $$1/2^i + 1/2^{i+1} + 1/2^{i+2} + ... $$ for some $i>1.$ It's not too hard to prove that such a sum converges to $1/2^{i-1}:$ \begin{equation}1/2^i + 1/2^{i+1} + 1/2^{i+2} + ... = 1/2^{i-1}.\end{equation} Substituting this into our expression for $S$ above we get \begin{eqnarray*} S & = & 2/2 & + & 3/4 & + & 4/8 & + & 5/16 & ... & + & n/2^{n-1} & + & ... \\ & = & 1 & & & & & & & & & & & \\ & + & 1 & & & & & & & & & & & \\ & + & 1/2 & & & & & & & & & & & \\ & + & 1/4 & & & & & & & & & & & \\ & + & 1/8 & & & & & & & & & & & \\ & + & ...& & & & & & & & & & & \\ \end{eqnarray*} In other words, \begin{eqnarray*} S & = & 1 + 1 + 1/2 + 1/4 + 1/8+ ... + 1/2^n + ... &\\ & = & 1 + 1 + 1 = 3\end{eqnarray*} Now, in reality, women are unable to have infinitely many children. Let's assume that the maximal number of children a woman can have is $m.$ The probability of a woman having $nReturn to the main article.