harmonic series 247

1

–

5

1

–

4

1

–

3

1

–

2

used because the nth harmonic produced by a violin

string is the tone produced by the string that is 1/n

times as long.

The corresponding

SERIES

for any harmonic

sequence necessarily diverges. In the study of

CONVERGENT

SERIES

, the comparison test shows that this must be the

case. (Compare a series of the form with

, which we know diverges.)

See also

HARMONIC SERIES

.

harmonic series The particular infinite sum 1 + +

+ + + … is called the harmonic

SERIES

. The

word harmonic is used because the nth harmonic pro-

duced by a violin string is the tone produced by the

string that is 1/ntimes as long.

Even though the terms of this series approach zero,

the series does not sum to a finite value. This can be

seen by grouping the terms of the series into sections of

length two, four, eight, 16, and so on, and making a

simple comparison:

That the series diverges means that summing suffi-

ciently many initial terms of the series will produce

answers arbitrarily large (although it may take a large

number of terms to do this). For example, summing

the first four terms produces an answer larger than 2,

the first 11 terms an answer larger than 3, the first

13,671 terms an answer larger than 10, and the first

1.53 ×1043 terms an answer larger than 100. Clearly

the series diverges to infinity very slowly.

In 1734 L

EONHARD

E

ULER

showed that, for large

values of n, the nth

PARTIAL SUM

of the harmonic series

can be well approximated by a

LOGARITHM

:

where γ≈0.577 is a constant (called E

ULER

’

S CON

-

STANT

) and the error in this approximation is no larger

than 1/n. (Notice that ln(n) + γ→∞as ngrows. This

again shows that the series diverges.)

The partial sums of the harmonic series are called

the harmonic numbers, and are denoted Hn. The first

10 harmonic numbers are:

One can use an induction argument to show that if

2k≤n< 2k+1, then the denominator of Hn(written in

reduced form) is a multiple of 2k. Consequently, no

denominator (except for the first) can be 1. This proves:

No harmonic number, other than the first, is

an integer.

The divergence of the harmonic series solves the amus-

ing rubber-band problem:

An infinitely tiny ant starts at one end of a rub-

ber band, 1 ft long, and crawls a distance of 1

in. toward the other end. It then pauses, and the

band is stretched 1 ft longer (to a total length of

2 ft), carrying the ant along with it to the 2-in.

position. The ant then crawls for another inch,

to the 3-in. position, and pauses while the band

is stretched another foot longer. This process of

walking an inch and pausing while the band

stretches a foot continues indefinitely. (We

assume the band is infinitely elastic.) Will the

ant ever make it to the end of the rubber band?