Let (an) and (bn)1 be two sequences. Assume thatthe sequence (1 ak)=1 is bounded, and that (bn)_₁ is decreasingn=with limit 0. Show thatΣanbnn=1nNis convergent. [HINT: Let An = Σ²_1 ªk. Then show that _1 anbn:=k=1N-1n=1ANON +1 An (bn — bn+1), and that Σn An(bn − bn+1) is absolutelyconvergent. For the last, you may need the Direct Comparison Test

Answered: Image /qna-images/answer/aa4ad2fa-7980-4ce9-94ab-571745123b89.jpg

Let (an) and (bn)1 be two sequences. Assume that the sequence (Σ1 ak)=1 is bounded, and that (bn) is decreasing n= with limit 0. Show that Σanbn n=1 n N k=1 is convergent. [HINT: Let An = Σ²_1 ªk. Then show that _1 anbn: ANON+1 An (bn — bn+1), and that Σn An (bn — bn+1) is absolutely convergent. For the last, you may need the Direct Comparison Test n=1 =

Let (an) and (bn)1 be two sequences. Assume that the sequence (Σ1 ak)=1 is bounded, and that (bn) is decreasing n= with limit 0. Show that Σanbn n=1 n N k=1 is convergent. [HINT: Let An = Σ²_1 ªk. Then show that _1 anbn: ANON+1 An (bn — bn+1), and that Σn An (bn — bn+1) is absolutely convergent. For the last, you may need the Direct Comparison Test n=1 =

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter10: Sequences, Series, And Probability

Section: Chapter Questions

Problem 22RE

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