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Show that the limit of ∑ 1/k = log (p/q) where the sum is from k = qn to pn

  1. For given integers p and q with 1 \leq q \leq p, prove

        \[ \lim_{n \to \infty} \sum_{k=qn}^{pn} \frac{1}{k} = \log \frac{p}{q}. \]

  2. Consider the series

        \[ 1 + \frac{1}{3} + \frac{1}{5} - \frac{1}{2} - \frac{1}{4} + \frac{1}{7} + \frac{1}{9} + \frac{1}{11} - \frac{1}{6} - \frac{1}{8} + \ + \ + \ - \ - \ \cdots. \]

    This is a rearrangement of the alternating harmonic series (\sum \frac{(-1)^n}{n}) in which there are three positive terms followed by two negative terms. Prove that the series converges and that the sum is equal to \log 2 + \frac{1}{2} \log \frac{3}{2}.


Incomplete.

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