$#*! I Jot On Napkins: Analysis

Sunday, February 28, 2010

Series are fun.

We learned in grade school that order in which one adds a bunch of numbers doesn't matter. So $\inline 1 + 2 - 3 = 2 - 3 + 1$ . This is not necessarily true when you are adding infinitely many numbers. Consider this series: $S = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \frac{1}{5} - \frac{1}{6} + \frac{1}{7} - \frac{1}{8} + \frac{1}{9} \cdots$ and this one: $S^{\prime} = 1 + \frac{1}{3} - \frac{1}{2} + \frac{1}{5} + \frac{1}{7} - \frac{1}{4} + \frac{1}{9} + \frac{1}{11} - \frac{1}{6} \cdots.$ The question is: are they equal?