Let a_{n} = ∑_{1}^{n} (-1)^{i+1}/i. Assume that lim_{n→∞} a_{n} = k. Rearrange the terms by taking two positive terms, then one negative term, then another two positive terms, then another negative term and so on. Let b_{n} be the sum of the first n terms of the rearranged series. Assume that lim_{n→∞} b_{n} = h. Show that b_{3n} = a_{4n} + a_{2n}/2, and hence that h ≠ k.

**Solution**

If we simply write out a_{4n} = (1 - 1/2 + 1/3 - 1/4 + ... - 1/4n) and a_{2n}/2 = (1/2 - 1/4 + 1/6 - 1/8 + ... - 1/4n) and add term by term we find that a_{4n} + a_{2n}/2 = b_{3n}.

Taking the limit, we have immediately that h = 3k/2. But clearly k > 0, since if we group the terms in pairs, each pair is positive. Hence h ≠ k.

© John Scholes

jscholes@kalva.demon.co.uk

26 Nov 1999