harmonicseries

The harmonic series is the infinite series sum from n=1 to infinity of 1/n. It is the sum of the reciprocals of the positive integers and is one of the simplest examples used to illustrate divergence in real analysis.

The series diverges: its partial sums S_N = sum from n=1 to N of 1/n grow without bound as N increases. Standard proofs use the integral test or Cauchy condensation. More precisely, the nth harmonic number H_N = sum_{n=1}^N 1/n satisfies H_N = log N + γ + o(1) as N tends to infinity, where γ is the Euler–Mascheroni constant.

Generalizations and related concepts include the generalized harmonic numbers H_N^(p) = sum_{n=1}^N 1/n^p. The corresponding infinite series converges for p > 1 and equals the Riemann zeta function ζ(p); for p = 1 the series diverges. The finite harmonic numbers H_N relate to the digamma function via H_N = ψ(N+1) + γ.

In broader analysis, the harmonic series is often cited as a standard example of a divergent series with very slow growth. By contrast, related alternating harmonic series sum_{n=1}^∞ (-1)^{n+1}/n converges to ln 2.

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