Alternating unit fractions/2 to 1 exchange/Convergence/Divergent reordering/Exercise

We consider the alternating series of the unit fractions ${}\sum _{n=1}^{\infty }x_{n}$ with

${}x_{n}=(-1)^{n+1}{\frac {1}{n}}\,,$

so

$1-{\frac {1}{2}}+{\frac {1}{3}}-{\frac {1}{4}}+{\frac {1}{5}}-{\frac {1}{6}}+{\frac {1}{7}}-{\frac {1}{8}}+{\frac {1}{9}}\cdots ,$

which converges.

a) Show that the reordered series

$1+{\frac {1}{3}}-{\frac {1}{2}}+{\frac {1}{5}}+{\frac {1}{7}}-{\frac {1}{4}}+{\frac {1}{9}}+{\frac {1}{11}}-{\frac {1}{6}}\cdots ,$

converges.

b) Give a reordering of the series which diverges.

Create a solution