Calculus II: Harmonic series

1. Harmonic series

A tone produced by a musical instrument as a guitar string has a frequency. The lowest frequency produced by any particular instrument is known as the fundamental frequency called the first harmonic of the instrument.

The sequence of musical tones whose frequencies are integral multiples of the frequency of the fundamental tone is called a harmonic frequencies. They are harmonic sequences.

If ƒ is the fundamental, the related harmonic sequence is:

ƒ, 2ƒ, 3ƒ, ..., nƒ, ...

The corresponding wavelength (which is inversely proportional to the frequency) sequence has the following form:

1/ƒ, 1/2ƒ, 1/3ƒ, ..., 1/nƒ, ...

or

1/ƒ {1, 1/2, 1/3, ..., 1/n, ...}

{1, 1/2, 1/3, ..., 1/n, ...} is an harmonic sequence, and

{1 + 1/2 + 1/3 + 1/4 + ...}    =	Σ (1/n) n: from 1 to ∞
is its corresponding harmonic series.

2. Divergence of harmonic series

The harmonic sequence is divergent. because:

lim (1/n) = 0
n → + ∞

The corresponding series diverge. Before proving this divergence by an integral, we first use the classical proof of Nicole Oresme (about 1350 in middle ages).

{1 + 1/2 + 1/3 + 1/4 + 1/5 + ... + 1/n + ... }

S₁ = 1
S₂ = 1 + 1(1/2)
S₄ = 1 + 1/2 + (1/3 + 1/4) > 1 + 1/2 + (1/4 + 1/4) = 1 + 2(1/2)
S₈ = 1 + 1/2 + (1/3 + 1/4) + (1/5 + 1/6 + 1/7 + 1/8) > 1 + 1/2 + (1/4 + 1/4) + (1/8 + 1/8 + 1/8 + 1/8) = 1 + 3(1/2)
...

S_2ⁿ ≥ 1 + n(1/2)

As lim (1 + n(1/2)) = + ∞
n → +∞

the subsequence {1 + n(1/2)} diverges, therefore the sequence {1/n} diverges.

Alternating harmonic series is convergent. Indeed:

{1 - 1/2 + 1/3 - 1/4 + 1/5 + ... + (- 1)^n-1/n + ... }

Let's justify this convergence: We have:

S₁ = 1
S₂ = 1 - 1/2
S₃ = 1 - 1/2 + 1/3
S₄ = 1 - 1/2 + 1/3 - 1/4
S₅ = 1 - 1/2 + 1/3 - 1/4 + 1/5

We remark that at each step, we subtract a large number (for even rank) and add a small number(for odd rank). Finally, we had subtracted a large number and added a small number to 1 which is the first term. Therefore the series converges to a number small than 1.