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Let 1 an be a convergent series with an 20 for all n. Prove that if there are constants bn converges. C> 0 and N> 0 such that for all n > N we have 0 ≤ b ≤ Can, then Let 0 0 be constants and let (bn) be a sequence such that 0≤ b, for all n and bn+1 ≤rb, for all n ≥ N. Prove that by+k ≤rby for all k>0. Use part (a) to conclude that b₁ converges.

Let 1 an be a convergent series with an 20 for all n. Prove that if there are constants bn converges. C> 0 and N> 0 such that for all n > N we have 0 ≤ b ≤ Can, then Let 0 0 be constants and let (bn) be a sequence such that 0≤ b, for all n and bn+1 ≤rb, for all n ≥ N. Prove that by+k ≤rby for all k>0. Use part (a) to conclude that b₁ converges.

College Algebra
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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part a and b

1. SERIES (a) Let an be a convergent series with an ≥0 for all n. Prove that if there are constants 1 bn converges. C> 0 and N> 0 such that for all n ≥ N we have 0 ≤ b ≤ Can, then (b) Let 0 <r<1 and N> 0 be constants and let (bn) be a sequence such that 0≤ b, for all n and bn+1 ≤rb, for all n ≥ N. Prove that by+k ≤rby for all k>0. Use part (a) to conclude that bn converges.
Transcribed Image Text:1. SERIES (a) Let an be a convergent series with an ≥0 for all n. Prove that if there are constants 1 bn converges. C> 0 and N> 0 such that for all n ≥ N we have 0 ≤ b ≤ Can, then (b) Let 0 <r<1 and N> 0 be constants and let (bn) be a sequence such that 0≤ b, for all n and bn+1 ≤rb, for all n ≥ N. Prove that by+k ≤rby for all k>0. Use part (a) to conclude that bn converges.
Expert Solution
Step 1

We know that a series ∑n=1∞an of non negative terms is convergent if and only if sequence of its partial sum is bounded above.

The sequence of partial sum sn of a series ∑n=1∞an is calculated as:

sn=a1+a2+⋯+an.

A series is convergent if the sequence of its partial sum is convergent.

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