The alternating harmonic series, while conditionally convergent, is not absolutely convergent: if the terms in the series are systematically rearranged, in general the sum becomes different and, dependent on the rearrangement, possibly even infinite. In particular, the sum is equal to the natural logarithm of 2: This series converges by the alternating series test. Is known as the alternating harmonic series. The first fourteen partial sums of the alternating harmonic series (black line segments) shown converging to the natural logarithm of 2 (red line). Leonhard Euler proved both this and also that the sum which includes only the reciprocals of primes also diverges, that is: Partial sums Where γ is the Euler–Mascheroni constant and ε k ~ 1 / 2 k which approaches 0 as k goes to infinity. This is because the partial sums of the series have logarithmic growth. For example, the sum of the first 10 43 terms is less than 100. The harmonic series diverges very slowly. The generalization of this argument is known as the integral test. Since this area is entirely contained within the rectangles, the total area of the rectangles must be infinite as well. The total area under the curve y = 1 / x from 1 to infinity is given by a divergent improper integral: Each rectangle is 1 unit wide and 1 / n units high, so the total area of the infinite number of rectangles is the sum of the harmonic series: Consider the arrangement of rectangles shown in the figure to the right. It is possible to prove that the harmonic series diverges by comparing its sum with an improper integral. It is still a standard proof taught in mathematics classes today. This proof, proposed by Nicole Oresme in around 1350, is considered to be a high point of medieval mathematics. More precisely, the comparison above proves that It follows (by the comparison test) that the sum of the harmonic series must be infinite as well. However, the sum of the second series is infinite: One way to prove divergence is to compare the harmonic series with another divergent series, where each denominator is replaced with the next-largest power of two:Įach term of the harmonic series is greater than or equal to the corresponding term of the second series, and therefore the sum of the harmonic series must be greater than or equal to the sum of the second series. There are several well-known proofs of the divergence of the harmonic series. In the Baroque period architects used them in the proportions of floor plans, elevations, and in the relationships between architectural details of churches and palaces. Harmonic sequences have been used by architects. Proofs were given in the 17th century by Pietro Mengoli, Johann Bernoulli, and Jacob Bernoulli. The exact value of this probability is given by the infinite cosine product integral C 2 divided by π.The fact that the harmonic series diverges was first proven in the 14th century by Nicole Oresme, but was forgotten. Schmuland's paper explains why this probability is so close to, but not exactly, 1 / 8. In particular, the probability density function of this random variable evaluated at +2 or at −2 takes on the value 0.124 999 999 999 999 999 999 999 999 999 999 999 999 999 764., differing from 1 / 8 by less than 10 −42. Byron Schmuland of the University of Alberta further examined the properties of the random harmonic series, and showed that the convergent series is a random variable with some interesting properties.
The fact of this convergence is an easy consequence of either the Kolmogorov three-series theorem or of the closely related Kolmogorov maximal inequality. Where the s n are independent, identically distributed random variables taking the values +1 and −1 with equal probability 1 / 2, is a well-known example in probability theory for a series of random variables that converges with probability 1. One way to prove divergence is to compare the harmonic series with another divergent series, where each denominator is replaced with the next-largest power of two:ġ + 1 2 + 1 3 + 1 4 + 1 5 + 1 6 + 1 7 + 1 8 + 1 9 + ⋯ ≥ 1 + 1 2 + 1 4 + 1 4 + 1 8 + 1 8 + 1 8 + 1 8 + 1 16 + ⋯ This was so particularly in the Baroque period, when architects used them to establish the proportions of floor plans, of elevations, and to establish harmonic relationships between both interior and exterior architectural details of churches and palaces. Historically, harmonic sequences have had a certain popularity with architects. The latter proof published and popularized by his brother Jacob Bernoulli. Proofs were given in the 17th century by Pietro Mengoli and by Johann Bernoulli, The divergence of the harmonic series was first proven in the 14th century by Nicole Oresme, but this achievement fell into obscurity.