University⏱ 4 min

Harmonic Series

1 + 1/2 + 1/3 + ... diverges

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H(n)ln(n)+γ
n = 30

H(30) = 3.9950

Sn=∑k=1n\ rac1k o∞S_n = \sum_{k=1}^n \ rac{1}{k} \ o \infty
Sn≈ln⁡n+γ,γ≈0.5772S_n \approx \ln n + \gamma,\quad \gamma \approx 0.5772

The harmonic series diverges, but extremely slowly – logarithmically. Even though terms go to 0, their sum goes to infinity.