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In mathematics , the harmonic series is the divergent infinite series. Every term of the series after the first is the harmonic mean of the neighboring terms; the phrase harmonic mean likewise derives from music.

The divergence of the harmonic series was first proven in the 14th century by Nicole Oresme , [1] but this achievement fell into obscurity.

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.

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

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:. 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. This proof, proposed by Nicole Oresme in around , is considered by many in the mathematical community [ by whom?

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

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. Since this area is entirely contained within the rectangles, the total area of the rectangles must be infinite as well.

More precisely, this proves that. The generalization of this argument is known as the integral test.

The harmonic series diverges very slowly. For example, the sum of the first 10 43 terms is less than In particular,. Leonhard Euler proved both this and also the more striking fact that the sum which includes only the reciprocals of primes also diverges, i.

The difference between H n and ln n converges to the Euler—Mascheroni constant. The difference between any two harmonic numbers is never an integer.

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

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We know that this series diverges. Thus the given series is divergent by the Comparison Test.

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Every term of the series after the first is the harmonic mean of the neighboring terms; the phrase harmonic mean likewise derives from music.

The divergence of the harmonic series was first proven in the 14th century by Nicole Oresme , [1] but this achievement fell into obscurity.

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.

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

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:. 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. This proof, proposed by Nicole Oresme in around , is considered by many in the mathematical community [ by whom?

It is still a standard proof taught in mathematics classes today. Cauchy's condensation test is a generalization of this argument. 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. Since this area is entirely contained within the rectangles, the total area of the rectangles must be infinite as well.

More precisely, this proves that. The generalization of this argument is known as the integral test. The harmonic series diverges very slowly.

For example, the sum of the first 10 43 terms is less than In particular,. Leonhard Euler proved both this and also the more striking fact that the sum which includes only the reciprocals of primes also diverges, i.

The difference between H n and ln n converges to the Euler—Mascheroni constant. The difference between any two harmonic numbers is never an integer.

This series converges by the alternating series test. In particular, the sum is equal to the natural logarithm of 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 :. This is known as the Leibniz series. A generalization of the harmonic series is the p -series or hyperharmonic series , defined as.

Related to the p -series is the ln-series , defined as. 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 [10] the properties of the random harmonic series, and showed that the convergent series is a random variable with some interesting properties.

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 The harmonic series can be counterintuitive to students first encountering it, because it is a divergent series even though the limit of the n th term as n goes to infinity is zero.

The divergence of the harmonic series is also the source of some apparent paradoxes. One example of these is the " worm on the rubber band ".

If the worm travels 1 centimeter per minute and the band stretches 1 meter per minute, will the worm ever reach the end of the rubber band?

The answer, counterintuitively, is "yes", for after n minutes, the ratio of the distance travelled by the worm to the total length of the rubber band is.

Because the series gets arbitrarily large as n becomes larger, eventually this ratio must exceed 1, which implies that the worm reaches the end of the rubber band.

However, the value of n at which this occurs must be extremely large: approximately e , a number exceeding 10 43 minutes 10 37 years.

Although the harmonic series does diverge, it does so very slowly. Another problem involving the harmonic series is the Jeep problem , which in one form asks how much total fuel is required for a jeep with a limited fuel-carrying capacity to cross a desert, possibly leaving fuel drops along the route.

The distance that can be traversed with a given amount of fuel is related to the partial sums of the harmonic series, which grow logarithmically.

And so the fuel required increases exponentially with the desired distance. Another example is the block-stacking problem : given a collection of identical dominoes, it is clearly possible to stack them at the edge of a table so that they hang over the edge of the table without falling.

The counterintuitive result is that one can stack them in such a way as to make the overhang arbitrarily large, provided there are enough dominoes.

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