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The harmonic series Σ k k=1 diverges to infinity.

The harmonic series Σ k k=1 diverges to infinity.

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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Could you please explain the following proof step by step, how do you approach a question like this, i feel like every question handles this a bit differently? 

Result 14.12 The harmonic series diverges to infinity. Proof k=1 k n 1 For n ≤ N, let s₂ = Σ. Thus, {s} is the sequence of partial sums for the harmonic k k=1 series. We show that lim S₁ = ∞. Let M be a positive integer and choose N = 22M. Let n→∞ n > N. Then, using Lemma 14.11, we have 1 Sn = 1+ + 2 = SN + 1 + + 1 N N + 1 + 1 N + 1 > SNS22M ≥ 1+ + N+2 2M 2 > M. + + 1 n + 1 n
Transcribed Image Text:Result 14.12 The harmonic series diverges to infinity. Proof k=1 k n 1 For n ≤ N, let s₂ = Σ. Thus, {s} is the sequence of partial sums for the harmonic k k=1 series. We show that lim S₁ = ∞. Let M be a positive integer and choose N = 22M. Let n→∞ n > N. Then, using Lemma 14.11, we have 1 Sn = 1+ + 2 = SN + 1 + + 1 N N + 1 + 1 N + 1 > SNS22M ≥ 1+ + N+2 2M 2 > M. + + 1 n + 1 n
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