Harmonic Sequences, Series & Harmonic Mean

Definition

The list 1, 1/2, 1/3, 1/4, 1/5, … looks orderly. Is it arithmetic, or geometric?

Those are the two families met so far: a constant step, or a constant ratio. One of them should fit.

Neither test passes: the differences keep shrinking and the ratios creep toward 1.

step	difference (arithmetic test)	ratio (geometric test)	reciprocals
1 → 1/2	−1/2	1/2	1 → 2
1/2 → 1/3	−1/6	2/3	2 → 3
1/3 → 1/4	−1/12	3/4	3 → 4
1/4 → 1/5	−1/20	4/5	4 → 5

The first two columns change at every step, so the list is neither arithmetic nor geometric — yet the last column is perfectly regular.

Flip every term and the pattern appears: the reciprocals 1, 2, 3, 4, 5 go up by 1 each time.

A harmonic sequence is neither arithmetic nor geometric. The differences between consecutive terms are not constant (they decrease), and the ratios are not constant either (they approach

1

from below). The defining property is that the reciprocals form an arithmetic progression — the structure lives one layer down.

The definition

A harmonic sequence is a sequence whose terms are the reciprocals of the terms of an arithmetic sequence.

A harmonic sequence is a sequence whose terms are the reciprocals of an arithmetic sequence. If

b_1, b_2, b_3, \ldots

is arithmetic with first term

b_1 > 0

and common difference

d > 0

, then the harmonic sequence is:

General Term (Harmonic Sequence)

a_n = \frac{1}{b_n} = \frac{1}{b_1 + (n-1)d}

Learn more about this formula: General Term (Harmonic Sequence) →

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So

1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \ldots

is harmonic: its reciprocals

1, 2, 3, 4, \ldots

form an arithmetic sequence with

d = 1

.

A harmonic sequence is an arithmetic sequence seen through reciprocals.

The simplest example takes

b_n = n

: the reciprocals

1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \ldots

form the harmonic sequence of natural number reciprocals. The sequence

\frac{1}{3}, \frac{1}{5}, \frac{1}{7}, \frac{1}{9}, \ldots

is also harmonic — it comes from the arithmetic sequence

3, 5, 7, 9, \ldots

with

b_1 = 3

and

d = 2

.

The common difference belongs to the denominators, not to the terms themselves.

Pitfall: Using a common difference of the terms

Treating

\frac{1}{3}, \frac{1}{5}, \frac{1}{7}, \ldots

as arithmetic with

d = \frac{1}{5} - \frac{1}{3} = -\frac{2}{15}

gives

a_{10} = \frac{1}{3} + 9 \cdot \left(-\frac{2}{15}\right) = -\frac{13}{15}

. The difference

d = 2

belongs to the denominators:

b_{10} = 3 + 9 \cdot 2 = 21

, so

a_{10} = \frac{1}{21}

.

Harmonic Sequence Notation

Notation

Harmonic Sequence Notation

Two letters for one sequence — the terms and the arithmetic layer under them — and one letter doing two different jobs.

The subscript as position, the brace form and sigma are set out at sequence notation;

\ln n

follows logarithm notation.

a_n = \dfrac{1}{b_n}

a sub n is one over b sub n

Two sequences on one page:

b_n

is the arithmetic one, with its own

b_1

and common difference

d

;

a_n

is the harmonic one built from it. The

d

belongs to the

b

's.

CasesThe

a

's themselves have no common difference:

\frac{1}{5} - \frac{1}{3} \neq \frac{1}{7} - \frac{1}{5}

.

Do not confuseApplying

d

to the terms directly — for

\frac{1}{3}, \frac{1}{5}, \frac{1}{7}, \ldots

it predicts

a_{10} = -\frac{13}{15}

instead of

\frac{1}{21}

.

H_n = \sum_{k=1}^{n} \dfrac{1}{k}

H sub n — the n-th harmonic number

A partial sum of the harmonic series. No closed form exists;

H_n \approx \ln n + \gamma

, with

\gamma \approx 0.5772

(section 3).

CasesThe full series

\sum_{n=1}^{\infty} \frac{1}{n}

diverges, so there is no finite

H_\infty

— unlike the geometric

S

.

Also written

1 + \frac{1}{2} + \cdots + \frac{1}{n}

.

Do not confuseThe harmonic mean

H

— same letter, no subscript.

H = \dfrac{n}{\frac{1}{a_1} + \cdots + \frac{1}{a_n}}

H, the harmonic mean

The reciprocal of the average of the reciprocals; for two numbers,

H = \frac{2ab}{a + b}

. Each inner term of a harmonic sequence is the harmonic mean of its neighbours (section 4).

CasesFor positive numbers

H \leq G \leq A

— see the geometric mean.

Do not confuse

H_n

. A subscript means a sum of reciprocals; no subscript means a mean.

All algebra symbols →↑ Sequences

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Notation · Cheat sheet

Sequence Notation at a Glance

The marks from the notation cards across the sequence pages: how each is read and what it means. Every group links to the full card on its page.

Sequences

a_n

a sub n

The term in place

n

; the subscript is a position.

a_1, a_2, a_3, \ldots

a one, a two, a three, and so on

The terms listed in order; the dots need a stated rule.

\{a_n\}_{n=1}^{\infty}

the sequence a sub n

The whole sequence as one object.

Arithmetic sequences

a_{n+1} = a_n + d

next term is this term plus d

The recurrence;

d

is the common difference,

a_1

the start.

\textstyle\sum_{k=1}^{n} a_k

sum of a sub k, k from 1 to n

Sigma: the sum

S_n

of the first

n

terms.

Geometric sequences

r

r, the common ratio

The factor from each term to the next:

a_n = a_1 r^{n-1}

.

S_n

S sub n

Sum of the first

n

terms;

n

counts terms.

S

S, the infinite sum

Equals

a_1 / (1 - r)

, only when

|r| < 1

.

G

G, the geometric mean

G = \sqrt{ab}

, the middle value of

a, G, b

.

Fibonacci sequence

F_n

F sub n

The

n

-th Fibonacci number, counted from

F_1 = F_2 = 1

.

\phi, \; \psi

phi and psi

The roots of

x^2 = x + 1

, about

1.618

and

-0.618

.

L_n

L sub n

Lucas numbers: the same rule, started at

1, 3

.

Harmonic sequencesThis page

a_n = 1/b_n

a sub n is one over b sub n

b_n

is the arithmetic sequence underneath.

H_n

H sub n

The harmonic number

1 + \frac{1}{2} + \cdots + \frac{1}{n}

.

H

H, the harmonic mean

For two numbers,

H = \frac{2ab}{a + b}

.

Triangular numbers

T_n

T sub n

The

n

-th triangular number,

\frac{n(n+1)}{2}

.

T_n = T_{n-1} + n

the one before plus a row of n

The recurrence, from

T_1 = 1

.

\binom{n+1}{2}

n plus one choose two

The same number, counted as pairs.

Square numbers

S_n = n^2

S sub n on this page

The

n

-th square: one term, not a sum.

S_n = S_{n-1} + 2n - 1

the one before plus the next odd number

The recurrence, from

S_1 = 1

.

\textstyle\sum_{k=1}^{n} k^2

sum of k squared

Square each term first, then add.

Prime numbers

p

p, a prime

The letter promises a prime.

p_1, \ldots, p_k

p sub one to p sub k

A list of primes, or the primes of one number.

\pi(n)

pi of n

How many primes are at most

n

.

Same letter, different job

S_n

A sum of

n

terms on the arithmetic and geometric pages; a single square on the square numbers page.

H_n

vs

H

With a subscript, a sum of reciprocals; without one, a mean.

\pi(n)

vs

\pi

A count of primes, not the circle constant.

\{a_n\}

An ordered sequence, not a set: terms keep their order and may repeat.

Sequence symbols table Sequences

Sequence Notation at a Glance·/algebra/sequencesLearn Math Class

Properties

Assuming the underlying arithmetic sequence has positive terms and positive common difference, the harmonic sequence is strictly decreasing: each term is smaller than its predecessor because the denominators grow.

The terms approach zero as

n \to \infty

, since the denominators grow without bound. However, the rate of decrease slows — the gap between

\frac{1}{n}

and

\frac{1}{n+1}

is

\frac{1}{n(n+1)}

, which itself shrinks as

n

increases.

THEOREM

Unlike arithmetic and geometric sequences, harmonic sequences have no closed-form expression for their partial sums.

The sum

\sum_{k=1}^{n} \frac{1}{k}

cannot be written as a simple formula in

n

. This absence is not a gap in technique — it is a fundamental property. The partial sums, known as harmonic numbers

H_n

, grow roughly as

\ln n

but involve no elementary closed form.

Fact: Never a whole number after

H_1

For every

n \geq 2

,

H_n

is not an integer. Over the least common denominator, every term except

\frac{1}{2^k}

, the largest power of

2

up to

n

, gets an even numerator, so the total numerator is odd while the denominator is even.

Written as reduced fractions, the first harmonic numbers are

H_1 = 1

,

H_2 = \frac{3}{2}

,

H_3 = \frac{11}{6}

,

H_4 = \frac{25}{12}

,

H_5 = \frac{137}{60}

and

H_6 = \frac{49}{20}

. The numerators and denominators follow no visible pattern, and they grow quickly: already

H_{10} = \frac{7381}{2520}

. Each value has to be built by adding one more fraction to the one before.

The Harmonic Series

The harmonic series is the infinite sum of reciprocals of the natural numbers:

The Harmonic Series

\sum_{n=1}^{\infty} \frac{1}{n} = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \cdots

THEOREM

This series diverges — its partial sums grow without bound, even though the individual terms shrink to zero.

The proof by grouping (attributed to Oresme, 14th century) is among the most elegant in mathematics. Group the terms after the first as follows:

The Harmonic Series

1 + \frac{1}{2} + \left(\frac{1}{3} + \frac{1}{4}\right) + \left(\frac{1}{5} + \frac{1}{6} + \frac{1}{7} + \frac{1}{8}\right) + \cdots

The first group contains

1

term summing to at least

\frac{1}{2}

. The next group contains

2

terms, each at least

\frac{1}{4}

, summing to at least

\frac{1}{2}

. The next contains

4

terms, each at least

\frac{1}{8}

, again summing to at least

\frac{1}{2}

. Each doubling block contributes at least

\frac{1}{2}

, and there are infinitely many blocks.

Fact: Drop every 9 and it converges

Remove each term whose denominator contains the digit

9

, and the sum stays below

80

(it is about

22.92

). There are

8 \cdot 9^{d-1}

such numbers with

d

digits, each term at most

\frac{1}{10^{d-1}}

, so the

d

-digit block adds at most

8 \cdot 0.9^{d-1}

, a geometric series with total

80

.

The harmonic series grows slowly — its partial sums approximate

\ln n + \gamma

, where

\gamma \approx 0.5772

is the Euler–Mascheroni constant. To reach a partial sum exceeding

10

, you need over

12{,}000

terms. But slowly is not the same as bounded: the sum eventually surpasses any finite threshold.

This stands in contrast to the geometric series

\sum \frac{1}{2^n}

, which converges to

1

. Both series have terms approaching zero, but geometric terms shrink fast enough (by a constant factor) to keep the total finite. Harmonic terms shrink too slowly.

RULE

The lesson: terms approaching zero is necessary for convergence but not sufficient.

Oresme’s blocks on the partial sums

The dots are the partial sums Hₙ for n = 1 to 16. The amber dashes are the grouping bounds: one block of terms brings the sum to at least 1.5 by n = 2, the next block to at least 2 by n = 4, then at least 2.5 by n = 8. Every doubling of n adds at least another 1/2, so the dots keep climbing past any level. The geometric sum 1/2 + 1/4 + 1/8 + … never passes the dashed line at 1.

The climb is slow: Hₙ grows like ln n, so passing 10 takes over 12,000 terms.

Harmonic Mean

The harmonic mean of

n

positive numbers

a_1, a_2, \ldots, a_n

is:

Harmonic Mean

H = \frac{n}{\frac{1}{a_1} + \frac{1}{a_2} + \cdots + \frac{1}{a_n}}

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For two numbers, this simplifies to:

Harmonic Mean

H = \frac{2ab}{a + b}

RULE

In a harmonic sequence, every term (except the first and last) is the harmonic mean of its two neighbors.

Pitfall: Averaging the terms instead of their reciprocals

Taking the term between

\frac{1}{3}

and

\frac{1}{7}

as their average gives

\frac{1}{2}\left(\frac{1}{3} + \frac{1}{7}\right) = \frac{5}{21}

, and

3, \frac{21}{5}, 7

is not arithmetic. Averaging the reciprocals

3

and

7

gives

5

, so the term is

\frac{1}{5}

.

The Mean Inequality

For any set of positive numbers, the three means satisfy:

AM-GM-HM Inequality

H \leq G \leq A

Learn more about this formula: AM-GM-HM Inequality →

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The harmonic mean is never greater than the geometric mean, which is never greater than the arithmetic mean. Equality throughout holds if and only if all numbers are identical.

For two positive numbers

a

and

b

:

The Mean Inequality

\frac{2ab}{a+b} \leq \sqrt{ab} \leq \frac{a+b}{2}

This chain links the three types of sequences through their associated means. Each mean captures a different notion of "average": arithmetic for additive processes, geometric for multiplicative processes, and harmonic for rates and reciprocals.

Pitfall: Averaging speeds with the arithmetic mean

Averaging

60

km/h out and

40

km/h back over the same distance as

\frac{60 + 40}{2} = 50

km/h is too high:

120

km each way takes

2 + 3 = 5

hours for

240

km, so

48

km/h. That is the harmonic mean

\frac{2 \cdot 60 \cdot 40}{60 + 40} = 48

.

Common Pitfalls

Four mistakes come up again and again with harmonic sequences, the harmonic series and the harmonic mean. Three of them are marked where they happen in the sections above; the fourth is the misreading that the section on the harmonic series answers. Here each is worked through in full, with the wrong and the right working side by side.

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1Using a common difference of the terms

The terms of a harmonic sequence are treated as an arithmetic sequence: a common difference is read off from the first two terms, and the formula

a_n = a_1 + (n-1)d

is applied to the terms themselves instead of to their reciprocals.

Start

For

\frac{1}{3}, \frac{1}{5}, \frac{1}{7}, \ldots

,

Wrong turn

Error here

d = \frac{1}{5} - \frac{1}{3} = -\frac{2}{15},

a_{10} = \frac{1}{3} + 9 \cdot \left(-\frac{2}{15}\right) = -\frac{13}{15}

Right path

The arithmetic sequence is the denominators

3, 5, 7, \ldots

, with

b_1 = 3

and

d = 2

.

b_{10} = 3 + 9 \cdot 2 = 21, \quad a_{10} = \frac{1}{b_{10}} = \frac{1}{21}

Why it happensThe general-term formula of an arithmetic sequence is the one most practised, and the difference is taken of whatever numbers are written down, not of their reciprocals.

How to checkCompute a second difference.

\frac{1}{7} - \frac{1}{5} = -\frac{2}{35}

, which is not

-\frac{2}{15}

, so the terms are not arithmetic. A negative answer is also impossible here: every term of this sequence is a positive fraction.

2Concluding convergence from shrinking terms

Because the terms

\frac{1}{n}

approach zero, the harmonic series is taken to add up to some finite total.

Wrong turn

Error here

\frac{1}{n} \to 0

, so the later additions become negligible and the partial sums settle at a limit.

Right path

Group the terms as in the grouping proof. The block from

\frac{1}{2^k + 1}

to

\frac{1}{2^{k+1}}

has

2^k

terms, each at least

\frac{1}{2^{k+1}}

, so it adds at least

\frac{1}{2}

:

H_{2^k} \geq 1 + \frac{k}{2}

The partial sums pass every bound, so the series diverges.

Why it happensFor the geometric series, terms that shrink to zero go together with a finite sum, and the condition that the terms approach zero gets mistaken for a test that settles convergence.

How to checkPut in a size. With

k = 20

the bound gives

H_{2^{20}} \geq 11

, so the first

1{,}048{,}576

terms already add up to more than

10

, and the next blocks keep adding at least

\frac{1}{2}

each.

3Averaging the terms instead of their reciprocals

The term between two neighbours in a harmonic sequence, or a harmonic mean between two values, is found by averaging the two values directly, as in an arithmetic sequence.

Start

Between

\frac{1}{3}

and

\frac{1}{7}

,

Wrong turn

Error here

\frac{1}{2}\left(\frac{1}{3} + \frac{1}{7}\right) = \frac{1}{2} \cdot \frac{10}{21} = \frac{5}{21}

Right path

Average the reciprocals

3

and

7

, then take the reciprocal back.

\frac{3 + 7}{2} = 5, \quad \text{so the term is } \frac{1}{5}

The two-number formula gives the same:

H = \frac{2 \cdot \frac{1}{3} \cdot \frac{1}{7}}{\frac{1}{3} + \frac{1}{7}} = \frac{\frac{2}{21}}{\frac{10}{21}} = \frac{1}{5}

.

Why it happensThe rule "a term is the average of its neighbours" is true for arithmetic sequences and is carried over without the step down to the reciprocals.

How to checkTake reciprocals of the three terms; they must form an arithmetic sequence.

3, 5, 7

does, with difference

2

. The wrong answer gives

3, \frac{21}{5}, 7

, with differences

\frac{6}{5}

and

\frac{14}{5}

.

4Averaging speeds with the arithmetic mean

Two speeds held over equal distances are averaged with the arithmetic mean, as if the trip spent equal time at each speed.

Start

60

km/h out and

40

km/h back over the same road,

Wrong turn

Error here

\frac{60 + 40}{2} = 50

Right path

Take the distance as

120

km each way. The trip out takes

\frac{120}{60} = 2

hours and the trip back

\frac{120}{40} = 3

hours, so

240

km take

5

hours:

\frac{240}{5} = 48 = \frac{2 \cdot 60 \cdot 40}{60 + 40}

The average speed is the harmonic mean,

48

km/h, whatever the distance.

Why it happensThe arithmetic mean is the everyday meaning of "average", and it is right when equal times are spent at each speed. With equal distances, more time is spent at the slower speed, which pulls the average down.

How to checkWork out the total time from the claimed average. At

50

km/h,

240

km would take

4.8

hours, but the two legs actually take

5

hours.

Common pitfalls with harmonic sequences·/algebra/sequences/harmonicLearn Math Class

Three of the four slips work with the terms where they should work with their reciprocals. A harmonic sequence is arithmetic only in its denominators, so differences, means and the next term are all found there, and an average speed over equal distances is the harmonic mean of the speeds. The fourth concerns the series: its terms shrink to

0

, yet the grouping argument shows that its partial sums pass every bound.

Harmonic Sequences FAQ

How can the harmonic series diverge when its terms shrink to zero?

+

Because terms shrinking to zero is necessary for convergence but not sufficient. The terms

\frac{1}{n}

do approach zero, yet they do so slowly enough that the running total still grows without bound. A geometric series with

|r| < 1

shrinks fast enough to converge; the harmonic series sits just on the wrong side of that threshold.Read more →

Is there a formula for $1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n}$ ?

+

No elementary one exists, and that is a genuine mathematical fact rather than a gap in technique. Unlike arithmetic and geometric sequences, harmonic partial sums have no closed form in

n

. They are called harmonic numbers, written

H_n

, and they grow roughly like

\ln n

, but no simple algebraic expression captures them exactly.Read more →

Which of the three means should you use?

+

It depends on what the quantities do. Arithmetic mean suits additive processes, geometric mean suits multiplicative ones such as growth rates, and harmonic mean suits rates and reciprocals, like averaging speeds over equal distances. They are ordered

H \leq G \leq A

for positive numbers, with all three equal only when every number is identical.Read more →

Is a harmonic number ever a whole number?

+

Only

H_1 = 1

; for every

n \geq 2

,

H_n

is not an integer. Put the terms over their least common denominator: the term

\frac{1}{2^k}

, where

2^k

is the largest power of

2

up to

n

, is the only one that gets an odd numerator, so the total numerator is odd while the denominator is even. The first values

\frac{3}{2}, \frac{11}{6}, \frac{25}{12}, \frac{137}{60}

all show it.Read more →

How many terms of the harmonic series does it take to pass 10?

+

More than

12{,}000

: the partial sums first exceed

10

at

n = 12{,}367

. Since

H_n

grows like

\ln n + \gamma

, with

\gamma \approx 0.5772

, reaching

10

needs

n

near

e^{10 - 0.5772}

, which is about

12{,}370

. Each further unit of sum multiplies the number of terms by about

e \approx 2.7

, yet the total still passes every bound in the end.Read more →

Does the harmonic series converge if every term with a 9 is removed?

+

Yes. Dropping each term whose denominator contains the digit

9

leaves a sum of about

22.92

. There are

8 \cdot 9^{d-1}

numbers with

d

digits and no

9

, each giving a term at most

\frac{1}{10^{d-1}}

, so the

d

-digit block adds at most

8 \cdot 0.9^{d-1}

. Those bounds form a geometric series with total

80

, which caps the whole sum. Almost every large number contains a

9

, so most terms are gone.Read more →

How is the geometric mean related to the arithmetic and harmonic means?

+

For two positive numbers it is exactly the geometric mean of the other two:

G^2 = A \cdot H

. Multiplying

A = \frac{a + b}{2}

by

H = \frac{2ab}{a + b}

cancels the

a + b

and leaves

ab = G^2

. At

a = 1

,

b = 4

the means are

A = 2.5

,

G = 2

and

H = 1.6

, and

2.5 \cdot 1.6 = 4 = 2^2

. For three or more numbers the identity fails in general.Read more →

Harmonic Sequences

Reciprocals of Arithmetic Progressions

Definition

Harmonic Sequence Notation

Harmonic Sequence Notation

Properties

The Harmonic Series

Harmonic Mean

The Mean Inequality

Common Pitfalls

Harmonic Sequences FAQ

How can the harmonic series diverge when its terms shrink to zero?

Is there a formula for 1+12+13+⋯+1n1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n}1+21​+31​+⋯+n1​?

Which of the three means should you use?

Is a harmonic number ever a whole number?

How many terms of the harmonic series does it take to pass 10?

Does the harmonic series converge if every term with a 9 is removed?

How is the geometric mean related to the arithmetic and harmonic means?

Is there a formula for $1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n}$ ?