We will now show that two different rearrangements of the alternating harmonic series add up to two different values.
(1) Consider[infinity]∑k=1(−1)k+1k= 1−12+13−14+15−16+17−···In the PreLab you showed that this series converged by satisfying the hypotheses of the Alternating Series Test. Thus, we may use the Remainder in Alternating Series (Theorem 8.20 in the text) Theorem to bound the actual sumS.

(a) Note that the first partial sum, S1= 1.
(i) The Remainder in Alternating Series Theorem states that the actual sumS(of which we will not determinethe exact value here) satisfies|S−S1|<.
(ii) Using properties of absolute values< S−S1<
(iii) AddingS1= 11−< S <1 +(iv) We concludeS <

(b) We repeat the steps forS2=
(i) the actual sumSsatisfies|S−S2|<.
(ii) Using properties of absolute values< S−S2<
(iii) So,< S <(iv) We concludeS