Harmonic series (mathematics)

Divergent sum of positive unit fractions

In mathematics, the harmonic series is the infinite series formed by summing all positive unit fractions: ∑ i = 1 ∞ 1 i = 1 1 + 1 2 + 1 3 + 1 4 + 1 5 + ⋯ . {\displaystyle \sum _{i=1}^{\infty }{\frac {1}{i}}={\frac {1}{1}}+{\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{4}}+{\frac {1}{5}}+\cdots .} The first n {\displaystyle n} terms of the series sum to approximately ln ⁡ n + γ {\displaystyle \ln n+\gamma } , where ln {\displaystyle \ln } is the natural logarithm and γ ≈ 0.577 {\displaystyle \gamma \approx 0.577} is the Euler–Mascheroni constant. Because the logarithm has arbitrarily large values, the harmonic series does not have a finite limit: it is a divergent series.

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