Putnam 1954 B2

Let $s$ denote the sum of the alternating harmonic series. Rearrange this series as follows $$1 + \frac{1}{3} - \frac{1}{2} + \frac{1}{5} +\frac{1}{7} - \frac{1}{4} + \frac{1}{9} + \frac{1}{11} - \ldots$$ Assume as known that this series converges as well and denote its sum by $S$ . Denote by $s_k, S_k$ respectively the $k$ -th partial sums of both series. Prove that $$\!\!\! \text{i}) S_{3n} = s_{4n} +\frac{1}{2} s_{2n}.$$ $$\text{ii}) S\ne s.$$