harmonic series

The harmonic series is the most famous cautionary tale in the subject: you add 1 + 1/2 + 1/3 + 1/4 + ..., and although each new term is tinier than the last and the terms march steadily to zero, the total still climbs without bound. It is the canonical proof that small terms are not enough for a series to converge.

Formally it is sum_{n=1}^infinity 1/n, the p-series with p = 1. The classic grouping argument of Oresme shows divergence: group the terms as 1, then 1/2, then (1/3 + 1/4), then (1/5 + ... + 1/8), and so on; each block sums to at least 1/2, so the partial sums exceed any bound. Equivalently the integral test applies, since the integral of 1/x from 1 to infinity is infinite.

The divergence is breathtakingly slow: the partial sum of the first N terms is approximately ln N + gamma, where gamma ≈ 0.5772 is the Euler–Mascheroni constant. To exceed 100 you need on the order of e^100 terms. This is the textbook example showing that the n-th term test is necessary but not sufficient — a_n -> 0 does not guarantee convergence.