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The harmonic series diverges, but slowly: Hₙ ≈ ln n + γ where γ ≈ 0.5772157 is the Euler-Mascheroni constant. The related p-series converges only when p > 1.
Harmonic number
Hₙ = Σₖ₌₁ⁿ 1/k ≈ ln n + γ + 1/(2n)
p-series
Σ 1/kᵖ converges if p > 1, diverges if p ≤ 1
About 5.1873775, and the asymptotic estimate gives 5.1873775 too — accurate to seven decimals.
Yes. Grouping terms into blocks each summing to at least ½ shows the total exceeds any bound, though it takes over 10⁴³ terms to pass 100.