Question

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1 1 1 + + (1) co 4 1 1 1 1 1 1 *82. (To illustrate that a conditionally convergent series, when rearranged, can have a different sum.) Consider 1- 2 and (2) 2 4 3 8 5 10 12 Check that (1) is conditionally convergent with limit S say, and that the terms in (2) are just a rear- rangement of those of (1). Denote the sum of the first n terms of (1) by Sn and of (2) by tn. Show that tzn = { $2n. (Hint: use induction. This implies that lim tzn IS and it is easily seen that lim t3n+1 = lim t3n+2 = lim t3n, so that the sum of (2) is į S. 1- + + +... = n-00 no n00 n00

Answer 1

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