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
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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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Step 1: Define the sequence of partial sums of an infinite series
VIEWStep 2: Write the sequence of partial sums of the series an
VIEWStep 3: Write the sequence of partial sums of the series anbn
VIEWStep 4: Use the infinite series concepts
VIEWStep 5: Use the theorem from step(1)
VIEWStep 6: Use the infinite series concepts and theorem ,to prove that the series anbn is convergent
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