Chapter 8.6, Problem 63E

Rearranging series It can be proved that if a series converges absolutely, then its terms may be summed in any order without changing the value of the series. However, if a series converges conditionally, then the value of the series depends on the order of summation. For example, the (conditionally convergent) alternating harmonic series has the value

$1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\cdots =1n2.$

Show that by rearranging the terms (so the sign pattern is + + −),

$1+\frac{1}{3}-\frac{1}{2}+\frac{1}{5}+\frac{1}{7}-\frac{1}{4}+\cdots =\frac{3}{2}1n2.$