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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 =

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Let (an) and (bn)1 be two sequences. Assume that the sequence (1 ak)=1 is bounded, and that (bn)_₁ is decreasing n= with limit 0. Show that Σanbn n=1 n N is convergent. [HINT: Let An = Σ²_1 ªk. Then show that _1 anbn: = k=1 N-1 n=1 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
Transcribed Image Text:Let (an) and (bn)1 be two sequences. Assume that the sequence (1 ak)=1 is bounded, and that (bn)_₁ is decreasing n= with limit 0. Show that Σanbn n=1 n N is convergent. [HINT: Let An = Σ²_1 ªk. Then show that _1 anbn: = k=1 N-1 n=1 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
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