harmonicserie

The harmonic series, denoted H, is the infinite series formed by summing the reciprocals of the positive integers: H = sum_{n=1}^∞ 1/n. It is the prototypical example of a series whose terms decrease toward zero, yet whose sum diverges, making it a foundational example in real analysis.

The series diverges. A standard elementary proof uses grouping: H = 1 + 1/2 + (1/3 + 1/4) + (1/5 + ... + 1/8) + (1/9 + ... + 1/16) + ... In each group, the sum is at least 1/2 (for example, 1/3 + 1/4 > 1/2, and the next group contains 2^k terms each at least 1/2^{k+1}, summing to at least 1/2). Since infinitely many groups contribute at least 1/2, the partial sums grow without bound. Equivalently, the integral test with ∫_1^∞ (1/x) dx diverges.

Related concepts include generalized harmonic numbers H_n = sum_{k=1}^n 1/k, and the broader p-series sum_{n=1}^∞ 1/n^p, which converges if p > 1 and diverges if p ≤ 1. The alternating harmonic series sum_{n=1}^∞ (-1)^{n+1}/n converges, in fact to ln 2.

Historically, the divergence of the harmonic series was analyzed as early as the medieval period and later given rigorous treatments by mathematicians such as Euler. The result illustrates that monotone decrease of terms to zero is not sufficient for convergence.