Harmonic series (mathematics)

In mathematics, the harmonic series is the divergent infinite series

: $\sum_^\infty\frac = 1 + \frac + \frac + \frac + \frac + \cdots.$

Its name derives from the concept of

overtone

s, or harmonics in music: the

wavelength

s of the overtones of a vibrating string are , , , etc., of the string's fundamental wavelength. Every term of the series after the first is the harmonic mean of the neighboring terms; the phrase ''harmonic mean'' likewise derives from music.

History

The divergence of the harmonic series was first proven in the 14th century by Nicole Oresme, but this achievement fell into obscurity. Proofs were given in the 17th century by

Pietro Mengoli

and by

Johann Bernoulli

,
the latter proof published and popularized by his brother Jacob Bernoulli.

Historically, harmonic sequences have had a certain popularity with architects. 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.

Divergence

There are several well-known proofs of the divergence of the harmonic series. A few of them are given below.

Comparison test

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:

:$\begin 1 & + \frac && + \frac && + \frac && + \frac && + \frac && + \frac && + \frac && + \frac && + \cdots \\$

pt

\geq 1 & + \frac && + \frac && + \frac && + \frac && + \frac && + \frac && + \frac && + \frac && + \cdots
\end

Each 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. However, the sum of the second series is infinite:

:$\begin & 1 + \left(\frac\right) + \left(\frac + \frac\right) + \left(\frac + \frac + \frac + \frac\right) + \left(\frac + \cdots + \frac\right) + \cdots \\$

pt

= & 1 + \frac + \frac + \frac + \frac + \cdots = \infty.
\end
(Here, "$=\infty$" is merely a notational convention to indicate that the partial sums of the series grow without bound.)

It follows (by the comparison test) that the sum of the harmonic series must be infinite as well. More precisely, the comparison above proves that

:$\sum_^ \frac \geq 1 + \frac$

for every positive integer .

This proof, proposed by Nicole Oresme in around 1350, is considered by many in the mathematical community to be a high point of

medieval mathematics

. It is still a standard proof taught in mathematics classes today. Cauchy's condensation test is a generalization of this argument.

Integral test

It is possible to prove that the harmonic series diverges by comparing its sum with an

improper integral

. Specifically, consider the arrangement of rectangles shown in the figure to the right. Each rectangle is 1 unit wide and units high, so the total area of the infinite number of rectangles is the sum of the harmonic series:

:$\begin \text \\ \text \end = 1 + \frac + \frac + \frac + \frac + \cdots$

Additionally, the total area under the curve from 1 to infinity is given by a divergent

improper integral

:

:$\begin \text \\ \text\end = \int_1^\infty\frac\,dx = \infty.$

Since this area is entirely contained within the rectangles, the total area of the rectangles must be infinite as well. More precisely, the first $k$ rectangles completely cover the region underneath the curve for $1 \leq x \leq k + 1$ and so

$\sum_^k \frac > \int_1^ \frac\,dx = \ln(k+1).$

The generalization of this argument is known as the integral test.

Rate of divergence

The harmonic series diverges very slowly. For example, the sum of the first 10⁴³ terms is less than 100. This is because the partial sums of the series have logarithmic growth. In particular,
:$\sum_^k\frac = \ln k + \gamma + \varepsilon_k \leq (\ln k) + 1$
where is the Euler–Mascheroni constant and which approaches 0 as goes to infinity.

Leonhard Euler

proved both this and also the more striking fact that the sum which includes only the reciprocals of primes also diverges, i.e.

:$\sum_\frac1p = \frac12 + \frac13 + \frac15 + \frac17 + \frac1 + \frac1 + \frac1 +\cdots = \infty.$

Partial sums

The finite partial sums of the diverging harmonic series,

: $H_n = \sum_^n \frac,$

are called harmonic numbers.

The difference between and converges to the Euler–Mascheroni constant. The difference between any two harmonic numbers is never an integer. No harmonic numbers are integers, except for .

Related series

Alternating harmonic series

The series

: $\sum_^\infty \frac = 1 - \frac + \frac - \frac + \frac - \cdots$

is known as the alternating harmonic series. This series converges by the alternating series test. In particular, the sum is equal to the natural logarithm of 2:

:$1 - \frac + \frac - \frac + \frac - \cdots = \ln 2.$

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.

The alternating harmonic series formula is a special case of the

Mercator series

, the Taylor series for the natural logarithm.

A related series can be derived from the Taylor series for the

arctangent

:

: $\sum_^\infty \frac = 1 - \frac + \frac - \frac + \cdots = \frac.$

This is known as the Leibniz series.

General harmonic series

The general harmonic series is of the form

:$\sum_^\frac ,$

where and are real numbers, and is not zero or a negative integer.

By the limit comparison test with the harmonic series, all general harmonic series also diverge.

-series

A generalization of the harmonic series is the -series (or hyperharmonic series), defined as

:$\sum_^\frac$

for any real number . When , the -series is the harmonic series, which diverges. Either the integral test or the Cauchy condensation test shows that the -series converges for all (in which case it is called the over-harmonic series) and diverges for all . If then the sum of the -series is , i.e., the

Riemann zeta function

evaluated at .

The problem of finding the sum for is called the Basel problem;

Leonhard Euler

showed it is . The value of the sum for is called

Apéry's constant

, since Roger Apéry proved that it is an irrational number.

ln-series

Related to the -series is the ln-series, defined as

:$\sum_^\frac$

for any positive real number . This can be shown by the integral test to diverge for but converge for all .

-series

For any

convex

, real-valued function such that

:$\limsup_\frac < \frac,$

the series

:$\sum_^\infty \varphi\left(\frac\right)$

is convergent.

Random harmonic series

The random harmonic series

:$\sum_^\frac,$

where the are independent, identically distributed random variables taking the values +1 and −1 with equal is a well-known example in probability theory for a series of random variables that converges with probability 1. The fact of this convergence is an easy consequence of either the Kolmogorov three-series theorem or of the closely related Kolmogorov maximal inequality. 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. In particular, the probability density function of this random variable evaluated at +2 or at −2 takes on the value ..., differing from by less than 10⁻⁴². Schmuland's paper explains why this probability is so close to, but not exactly, . The exact value of this probability is given by the infinite cosine product integral divided by .

Depleted harmonic series

The depleted harmonic series where all of the terms in which the digit 9 appears anywhere in the denominator are removed can be shown to converge to the value . In fact, when all the terms containing any particular string of digits (in any base) are removed, the series converges.

Applications

The harmonic series can be counterintuitive