THE HARMONIC SERIES

The harmonic series is the infinite series ∞ n=1 1 n = 1 + 1 2 + 1 3 + 1 4 + · · · The series diverges to infinity even though lim n→∞ 1/n = 0. The divergence of the harmonic series was first proved by the medieval French scholar Nicolas d'Oresme (1323–1382). His proof is the following. Since the series 1 + 1 2 + 1 2 + 1 2 + · · · diverges, the harmonic series diverges as well. We can also prove that the harmonic series diverges using integration. 0 x y 1 1 2 3 4. .. n n + 1 y = 1/x The sum of the area of the shaded rectangles is larger than the area between the x-axis and the graph of y = 1/x over 1 ≤ x ≤ n + 1. Therefore, ln(n + 1) = n+1 1 1 x dx ≤ 1 + 1 2 + 1 3 + · · · + 1 n. Since lim n→∞ ln(n + 1) = ∞, we conclude that the harmonic series diverges.