H(n) − ln(n) converges to the Euler-Mascheroni constant γ
The difference between the harmonic sum and ln(n) approaches γ ≈ 0.5772 as n → ∞. Convergence is very slow – the gap is still 0.001 at n = 1000.
Key facts about γ
γ = lim(n→∞) [H(n) − ln(n)] ≈ 0.5772156649…
γ = −Γ'(1) = −∫₀^∞ e⁻ˣ ln(x) dx
Whether γ is irasional is tidak diketahui – one of the oldest open problems in mathematics.
Harmonic staircase H(n) versus smooth ln(n) + γ
The harmonic partial sums H(n) (red, stepped) versus ln(n)+γ (blue, smooth). The gap between them approaches 0 but oscillates: H(n)−ln(n) → γ.
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Pertanyaan
Bagaimana gamma didefinisikan menggunakan deret harmonik?
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Browse the digits of Euler-Mascheroni Constant γ
γ has no final digit
Euler-Mascheroni Constant γ is irrational. Its decimal expansion never ends and never repeats. The digits shown below are verified against the harmonic-logarithm limit.
γ = lim(n→∞) (1 + 1/2 +... + 1/n − ln n)
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