Table of Contents
- What is a Harmonic Series?
- Harmonic Number
- Harmonic Series Formula
- Harmonic Series Diverges
- Harmonic Series Examples
What is a harmonic series? Here is the series:
This is the harmonic series math definition. This is the series of rational numbers with numerator one and integer denominator values that progressively increase.
The harmonic series is useful throughout mathematics due to its unique properties:
These properties will be explored later in this lesson.
The harmonic series is most commonly known for its usefulness in music. Since the harmonic series is the only natural scale, it is used as the basis for all tone systems. So, whenever a tone sounds, overtones oscillate with it. However, the sound is simultaneous. This structure is always the same and corresponds to a harmonic series that arises in mathematics. It is difficult to hear the harmonics since they vibrate at the same time as the chord so they sound like a single tone.
Here is an image in Figure 1 showing the harmonic partials (harmonic partial sums) that arise in music as they appear on a music sheet:
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The harmonic series is also useful in mathematics for a wide variety of processes. The most important is probably the counterexample to the claim that '"if the limit of the terms in a series is zero, then the series converges."' Clearly, the limit of the terms of this series is zero but, as will be shown later, this series does not converge.
What is a harmonic number? Well, a harmonic number is any number that can be represented as a partial sum of the harmonic series. That is, a harmonic number, {eq}H_n {/eq}, is the nth partial sum of the harmonic series:
So, if the harmonic series is stopped at a certain n-value, then the resulting sum is the nth partial sum and a harmonic number.
The final term in the nth partial sum is called the nth term. Since the nth partial sum starts at the first term and ends at the nth term, the resulting sum is called the nth partial sum. Moreover, the sum is called a partial sum because it does not take into account every single term of the series but just a portion of it.
Here are some examples of harmonic numbers:
These are all harmonic numbers since they can be represented as partial sums of the harmonic series.
The harmonic series formula is:
Now, what is the sum of a harmonic series? It is possible to simply sum the terms of the partial sum. Unfortunately, unlike other series, there is no simple formula for calculating the nth partial sum of a harmonic series. The only way to find the partial sum of a harmonic series is to simply sum the terms of the partial sum:
This formula will work for any partial sum of a harmonic series. Ultimately, the sum of harmonic series is given by the sum of the terms of the partial series up to the nth term.
A series, {eq}\sum_{n=1}^{\infty} a_n {/eq}, with nth partial sums denoted by {eq}s_n = \sum_{k=1}^{n} a_k {/eq} can be convergent or divergent:
Now, does the harmonic series diverge or does the harmonic series converge? What is the harmonic series convergence? Well, here is the classical proof used by French scholar Nicole Oresme to show that the harmonic series diverges:
Proof:
So, this has shown that the harmonic series is divergent. But, is there another way? Here is another proof using elementary calculus:
Proof:
Here is an image in Figure 2 showing this improper integral and the area under the curve:
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In the image in Figure 2, the graph of {eq}\frac{1}{x} {/eq} is shown along with the area under its curve from one to infinity. Since the area under the curve is infinite over this interval, the series associated to this function must also diverge.
Here are some harmonic series examples:
Example 1:
What is the fourth partial sum of the harmonic sequence?
Solution:
Example 2:
A p-series is a series of the form {eq}\sum_{n=1}^{\infty} \frac{1}{n^p} {/eq} where p is any positive real number. At what values of p does the p-series diverge?
Solution:
Example 3:
Consider the alternating harmonic series: {eq}\sum_{n=1}^{\infty} \frac{(-1)^{n}}{n} {/eq}. This is the harmonic series whose terms are alternating signs. Does this series converge or diverge? If it converges, what does it converge to?
Solution:
The harmonic series is the series: {eq}\sum_{n=1}^{\infty} \frac{1}{n} {/eq}. The harmonic series is a useful series that arises in mathematics and it has special usefulness in music since the harmonic series forms a basis of all tone systems. Now, a harmonic number is a number that can be represented as a partial sum of the harmonic series. That is if the harmonic series is taken only to a finite number of values, whatever number this sums to is called a harmonic number. Unfortunately, there is no easy way to sum a harmonic series. The only way to effectively sum a partial harmonic series is to add the terms that arise in the summation.
Finally, the series can either converge or diverge. A series is convergent if its sequence representation is convergent and its nth partial sum converges to a real number. If either of these conditions fail, then the series is said to be divergent. If a series is divergent, then it either has no real number output or it goes to infinity rather than a real number. It can be shown that the harmonic series is divergent using classical methods or methods that arise in calculus, such as the integral test.
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The harmonic series is the series starting at 1 and going to infinity of 1/n. It looks like 1 + 1/2 + 1/3 + 1/4 +.... It is the series of rational numbers whose numerators are one and whose denominators are integers increasing by one each time.
The harmonic series does not converge absolutely. Simply take the absolute value of 1/n and the output is still 1/n. Therefore, the absolute value of the harmonic series is the harmonic series which diverges.
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