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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.

B2