2 2. Consider the alternating series 1 –+-+.i.e. Ear where ar = (-1)k-1! for all k e N.Let SnE- ak denote the partial sums.(a) Show that ak is not absolutely convergent.(b) Show that for all n E N,n1S2nand S2n+11= S2n +2 2k(2k – 1)2n + 1k=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 E, ak = S.

S = 1-12+ 13- 14+ ..... = ∑kak where ak = -1k-11k ; k∈ℕ.Sn =∑k=1nak is Partial sum.…

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.

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.

College Algebra

7th Edition

ISBN:9781305115545

Author:James Stewart, Lothar Redlin, Saleem Watson

Publisher:James Stewart, Lothar Redlin, Saleem Watson

Chapter8: Sequences And Series

Section8.3: Geometric Sequences

Problem 4E: (a) The nth partial sum of a geometric sequence an=arn1 is given by Sn=. (b) The series...

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