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(c) Let a be a positive real number and let b₁ = for all n. Prove that bn converges. 2=1 (d) Let x be a real number and let b₁ = for all n. Prove that be converges. n! Zn=1

(c) Let a be a positive real number and let b₁ = for all n. Prove that bn converges. 2=1 (d) Let x be a real number and let b₁ = for all n. Prove that be converges. n! Zn=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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PART C AND PART D (asked part a and b in another question)

1. SERIES (a) Let 1 an be a convergent series with an ≥0 for all n. Prove that if there are constants C> 0 and N> 0 such that for all n ≥ N we have 0 ≤ b ≤ Can, then 1 bn converges. (b) Let 0<r<1 and N> 0 be constants and let (bn) be a sequence such that 0 ≤ bn for all n and bn+1 ≤rbn for all n > N. Prove that by+k ≤rby for all k>0. Use part (a) to conclude that b₁ converges. = (c) Let a be a positive real number and let bn for all n. Prove that bn converges. (d) Let a be a real number and let b₁ = for all n. Prove that be converges. n=1
Transcribed Image Text:1. SERIES (a) Let 1 an be a convergent series with an ≥0 for all n. Prove that if there are constants C> 0 and N> 0 such that for all n ≥ N we have 0 ≤ b ≤ Can, then 1 bn converges. (b) Let 0<r<1 and N> 0 be constants and let (bn) be a sequence such that 0 ≤ bn for all n and bn+1 ≤rbn for all n > N. Prove that by+k ≤rby for all k>0. Use part (a) to conclude that b₁ converges. = (c) Let a be a positive real number and let bn for all n. Prove that bn converges. (d) Let a be a real number and let b₁ = for all n. Prove that be converges. n=1
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