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Consider the recursively defined sequence (Hn)n>1 where 1. H₁ = 1 2. Hn+1 = Hn+n1 for n ≥ 1 > Compare the quantities H₂n+1 and H2n + 1/2 for n ≥ 0 by creating a table with small values of n. Prove your relation by induction.

Consider the recursively defined sequence (Hn)n>1 where 1. H₁ = 1 2. Hn+1 = Hn+n1 for n ≥ 1 > Compare the quantities H₂n+1 and H2n + 1/2 for n ≥ 0 by creating a table with small values of n. Prove your relation by induction.

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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Consider the recursively defined sequence \((H_n)_{n \geq 1}\) where: 1. \(H_1 = 1\) 2. \(H_{n+1} = H_n + \frac{1}{n+1}\) for \(n \geq 1\) Compare the quantities \(H_{2n+1}\) and \(H_{2n} + 1/2\) for \(n \geq 0\) by creating a table with small values of \(n\). Prove your relation by induction.
Transcribed Image Text:Consider the recursively defined sequence \((H_n)_{n \geq 1}\) where: 1. \(H_1 = 1\) 2. \(H_{n+1} = H_n + \frac{1}{n+1}\) for \(n \geq 1\) Compare the quantities \(H_{2n+1}\) and \(H_{2n} + 1/2\) for \(n \geq 0\) by creating a table with small values of \(n\). Prove your relation by induction.
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