The harmonious sequence

The world is full of sequences. From fractals to the finances. But, I would like to show you one particular sequence. In order to understand the world, we need to understand this one. That's why I'm going with today's topic: the harmonious sequence.

The blog will be divided into three sections. In the first section the harmonic sequence will be presented, then I will talk about the history of this sequence and at the end of the section the mathematical representation will be shown. The second section presents the recursive and explicit formula of it and explains the properties of both, then I will explain the convergence of it using a graph and finally, I will explain the superficial use of the sequence. In the third section, I will explain how it is used in physics, finance, and music. Now let's get started with the blog!

Basic knowledge

The harmonic sequence is a mathematical sequence in which the reciprocals of natural numbers occur as consecutive members. The general formula for a_n is a_n=1/n.

This sequence has a long history dating back to antiquity. The Greeks were also interested in the harmonic sequence, as in the book: “Plato and the sequences”. Other scientists, such as Nicholas of Oresme (French preacher, mathematician, bishop, physicist, astronomer and scientist, as well as philosopher of the century) also dealt with the sequence. The modern development and use of the harmonic series spans various fields such as mathematics, physics, engineering, and finance.

The harmonic series can be represented mathematically by the general formula a_n=1/n, where n represents the position of the member in the series.

Actually calculating the sequence is easy, but it can be used in many ways. The sequence goes: a_1 =1; a_2=½; a_3=⅓; a_4=¼; and so on.

Mathematic manner?

The recursive sequence, the harmonic number series, applies: a_1= 1 and a_n +1 = a_n + 1/(n+1). This recursive sequence shows how each member of the sequence is determined by the previous member and an additional reciprocal of the current index.

The explicit sequence applies: a_n= 1/n. If one of these two is used to calculate the order of the sequence members, this result is obtained:

n 1 2 3 4 5 6 7 8 9

a_n

1/2

1/3

1/4

1/5

1/6

1/7

1/8

1/9

There are different ways to check the divergence of this sequence. The first way could already be done with these values. Let's put the sequence elements into a graph:

Here it becomes clear that the graph converges to zero.

Another way is to do it mathematically. We use the general, explicit formula. In this we insert further larger numbers:

a_n=1/n; a_{1}=1; a_{10}=1/10; a_{100}=1/100; a_{1000}=1/1000; a_{10000}=1/10000

As can be seen here, the higher the sequence element, the smaller the result. Written in a mathematical way:

lim a_n =0

n → ∞

Let’s see the graph again:

We have already established that this converges to zero, so it has a zero asymptote. But what does that mean? It simply means that the sequence members can continue to increase, and the result will never be less than zero. This asymptote actually shows us something else: the monotonicity of the graph. This graph is decreasing, as we can clearly see, but since it has an asymptote, the graph is only monotonically decreasing, not strictly monotonic.

The harmonic sequence in life?

The harmonic series has other applications that are related to mathematics, but also apply to other subjects. For example, in physics it is used to model periodic movements such as pendulum swings, spring movements and sound waves. These periods can therefore be described using the harmonic series.

In financial mathematics, the harmonic series is used to calculate average returns. This is important for analyzing investments and portfolio performance.

Debbie: Did you know that music also uses the harmonic series? This is where the harmonic series is used in the musical scale and harmony theory to determine the intervals between consecutive notes. The relationship between the frequencies of tones can be described by the harmonic series.

Although I can continue with the harmonic sequence and its applications, it's time to call it a day… I hope that the basic info of harmonic sequence is now clear to you. Think about where else you can find this episode, as it can be seen in other places as well. Bye, see you next time!

Sources of reference:

P. D. I'm sorry if the text is not well written, English it's not my first language.

By your trusted math teacher, Len.

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