2. Consider the alternating series 1 –+-+..., i.e. Eak where az = (-1)k-1 for all k e N. 4 Let Sn =E- ak denote the partial sums. (a) Show that Eak is not absolutely convergent. (b) Show that for all n E N, 1 San =E 1 = S2n + and S2n+1 2k(2k – 1) 2n + 1 k=1 (Hint: group pairs of odd and even terms.) (c) Prove that the sequence of even partial sums (S2n) has a limit S. (d) Prove that the whole sequence (Sm) has the same limit S, and hence , ak = S.

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