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Harmonic number

Harmonic number is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Harmonic number rather than just read about it. In short: In mathematics, the n-th harmonic number is the sum of the reciprocals of the first n natural numbers: H n = 1 + 1 2 + 1 3 + ⋯ + 1 n = ∑ k = 1 n 1 k . {\displaystyle H_{n}=1+{\frac {1}{2}}+{\frac {1}{3}}+\cdots +{\frac {1}{n}}=\sum _{k=1}^{n}{\frac {1}{k}}.} Starting from n = 1, the sequence of harmonic numbers begins: 1 , 3 2 , 11 6 , 25 12 , 137 60 , … {\displaystyle 1,{\frac {3}{2}},{\frac {11}{6}},{\frac {25}{12…

Harmonic number — main illustration
Harmonic number — illustration

Key takeaways

  • Harmonic number belongs to mathematics; place it in that map before memorising details.
  • Learn the definition first, then one example that makes the definition concrete.
  • Connect Harmonic number to a quantity you can measure, compute or draw — that is where exam questions come from.
  • Reproduce the core statement of Harmonic number from memory before moving on to harder problems.

Reference excerpt

In mathematics, the n-th harmonic number is the sum of the reciprocals of the first n natural numbers:

H n = 1 + 1 2 + 1 3 + ⋯ + 1 n = ∑ k = 1 n 1 k . {\displaystyle H_{n}=1+{\frac {1}{2}}+{\frac {1}{3}}+\cdots +{\frac {1}{n}}=\sum _{k=1}^{n}{\frac {1}{k}}.}

Starting from n = 1, the sequence of harmonic numbers begins:

1 , 3 2 , 11 6 , 25 12 , 137 60 , … {\displaystyle 1,{\frac {3}{2}},{\frac {11}{6}},{\frac {25}{12}},{\frac {137}{60}},\dots }

Harmonic numbers are related to the harmonic mean in that the n-th harmonic number is also n times the reciprocal of the harmonic mean of the first n positive integers. Harmonic numbers have been studied since antiquity and are important in various branches of number theory. They are sometimes loosely termed harmonic series, are closely related to the Riemann zeta function, and appear in the expressions of various special functions. The harmonic numbers roughly approximate the natural logarithm function and thus the associated harmonic series grows without limit, albeit slowly. In 1737, Leonhard Euler used the divergence of the harmonic series to provide a new proof of the infinity of prime numbers. His work was extended into the complex plane by Bernhard Riemann in 1859, leading directly to the celebrated Riemann hypothesis about the distribution of prime numbers. When the value of a large quantity of items has a Zipf's law distribution, the total value of the n most-valuable items is proportional to the n-th harmonic number. This leads to a variety of surprising conclusions regarding the long tail and the theory of network value. Counting the number of factors 2 in the numerator and denominator of the harmonic numbers written as a fraction shows that, except for the case n = 1, the harmonic numbers have even denominators and thus are never integers. The Bertrand-Chebyshev theorem can also be used to show that the denominator of the nth harmonic number is divisible by the largest prime not larger than n.

List

Identities involving harmonic numbers By definition, the harmonic numbers satisfy the recurrence relation

H n + 1 = H n + 1 n + 1 . {\displaystyle H_{n+1}=H_{n}+{\frac {1}{n+1}}.}

The harmonic numbers are connected to the Stirling numbers of the first kind by the relation

H n = 1 n ! [ n + 1 2 ] . {\displaystyle H_{n}={\frac {1}{n!}}\left[{n+1 \atop 2}\right].}

The harmonic numbers satisfy the series identities

∑ k = 1 n H k = ( n + 1 ) H n − n {\displaystyle \sum _{k=1}^{n}H_{k}=(n+1)H_{n}-n}

and

∑ k = 1 n H k 2 = ( n + 1 ) H n 2 − ( 2 n + 1 ) H n + 2 n . {\displaystyle \sum _{k=1}^{n}H_{k}^{2}=(n+1)H_{n}^{2}-(2n+1)H_{n}+2n.}

These two results are closely analogous to the corresponding integral results

∫ 0 x log ⁡ y d y = x log ⁡ x − x {\displaystyle \int _{0}^{x}\log y\ dy=x\log x-x}

and

∫ 0 x ( log ⁡ y ) 2 d y = x ( log ⁡ x ) 2 − 2 x log ⁡ x + 2 x . {\displaystyle \int _{0}^{x}(\log y)^{2}\ dy=x(\log x)^{2}-2x\log x+2x.}

Identities involving π There are several infinite summations involving harmonic numbers and powers of π:

… excerpt ends here. Continue reading the full article.

Illustrations

Harmonic number: The harmonic number H n {\displaystyle H_{n}} with n = ⌊ x ⌋ {\displaystyle n=\lfloor x\rfloor } (red line) with its asymptotic limit γ + ln ⁡ ( x ) {\displaystyle \gamma +\ln(x)} (blue line) where γ {\displaystyle \gamma } is the Euler–Mascheroni constant.
The harmonic number H n {\displaystyle H_{n}} with n = ⌊ x ⌋ {\displaystyle n=\lfloor x\rfloor } (red line) with its asymptotic limit γ + ln ⁡ ( x ) {\displaystyle \gamma +\ln(x)} (blue line) where γ {\displaystyle \gamma } is the Euler–Mascheroni constant.
Harmonic number: Graph demonstrating a connection between harmonic numbers and the natural logarithm. The harmonic number Hn can be interpreted as a Riemann sum of the integral: ∫ 1 n + 1 d x x = ln ⁡ ( n + 1 ) . {\displaystyle \int _{1}^{n+1}{\frac {dx}{x}}=\ln(n+1).}
Graph demonstrating a connection between harmonic numbers and the natural logarithm. The harmonic number Hn can be interpreted as a Riemann sum of the integral: ∫ 1 n + 1 d x x = ln ⁡ ( n + 1 ) . {\displaystyle \int _{1}^{n+1}{\frac {dx}{x}}=\ln(n+1).}

Worked examples

Example 1 — a first encounter with Harmonic number

Start with the simplest possible case. Write down what Harmonic number claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Harmonic number before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Harmonic number ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Harmonic number

In research
Harmonic number appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry
Engineering practice reuses Harmonic number in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom
Harmonic number is common in secondary-school and first-year university syllabi. It links to neighbouring topics Number theory, so understanding it makes those chapters shorter.
In everyday life
Look for Harmonic number outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.
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How to study Harmonic number in 20 minutes

  1. Read the reference excerpt below once, without taking notes.
  2. Close the page and write down what Harmonic number means in your own words.
  3. Compare your version with the excerpt and mark what you missed.
  4. Work through the three examples above with pen and paper.
  5. Explain Harmonic number out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Harmonic number in simple terms?

In mathematics, the n-th harmonic number is the sum of the reciprocals of the first n natural numbers: H n = 1 + 1 2 + 1 3 + ⋯ + 1 n = ∑ k = 1 n 1 k . {\displaystyle H_{n}=1+{\frac {1}{2}}+{\frac {1}{3}}+\cdots +{\frac {1}{n}}=\sum _{k=1}^{n}{\frac {1}{k}}.} Starting from n = 1, the sequence of har…

Why does Harmonic number matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Harmonic number?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Harmonic number.

Tags

  • Number theory

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