[infinity]
The famous Harmonic Series S = Σ 1/n is known to diverge. In class, we proved this using the integral test.
ⁿ⁼¹
Work through the steps below to build a proof that needs no calculus at all. Start with the partial-sum notation
ₙ
SN = Σ 1/n N ∈ N.
ⁿ⁼¹
(b) Explain why S₂ₙ₊₁ - S₂ₙ ≥ 1/2 for each integer n ≥ 0.
Suggestion: Show your readers a pattern by expressing a few sample differences, like S2-S1, S4-S2, S8-S4, etc., in a natural format that makes the requested inequality easy to justify.
(c) Combine your findings in part (b) with the limit-definition for the value of S to explain why S = +[infinity].