Difference between revisions of "Harmonic sequence"
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In [[algebra]], a '''harmonic sequence''', sometimes called a '''harmonic progression''', is a [[sequence]] of numbers such that the difference between the reciprocals of any two consecutive terms is constant. In other words, a harmonic sequence is formed by taking the reciprocals of every term in an [[arithmetic sequence]]. | In [[algebra]], a '''harmonic sequence''', sometimes called a '''harmonic progression''', is a [[sequence]] of numbers such that the difference between the reciprocals of any two consecutive terms is constant. In other words, a harmonic sequence is formed by taking the reciprocals of every term in an [[arithmetic sequence]]. | ||
− | For example, <math>1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}</math> and <math> | + | For example, <math>-1, -\frac{1}{2}, -\frac{1}{3}, -\frac{1}{4}</math> and <math>6, 3, 2, \frac{6}{4}</math> are harmonic sequences; however, <math>0, \frac{1}{3}, \frac{1}{6}, \frac{1}{9}</math> and <math>\frac{1}{4}, \frac{1}{12}, \frac{1}{36}, \frac{1}{108}, \ldots</math> are not. |
More formally, a harmonic progression <math>a_1, a_2, \ldots , a_n</math> biconditionally satisfies <math>1/a_2 - 1/a_1 = 1/a_3 - 1/a_2 = \cdots = 1/a_n - 1/a_{n-1}.</math> A similar definition holds for infinite harmonic sequences. It appears most frequently in its three-term form: namely, that constants <math>a</math>, <math>b</math>, and <math>c</math> are in harmonic progression if and only if <math>1/b - 1/a = 1/c - 1/b</math>. | More formally, a harmonic progression <math>a_1, a_2, \ldots , a_n</math> biconditionally satisfies <math>1/a_2 - 1/a_1 = 1/a_3 - 1/a_2 = \cdots = 1/a_n - 1/a_{n-1}.</math> A similar definition holds for infinite harmonic sequences. It appears most frequently in its three-term form: namely, that constants <math>a</math>, <math>b</math>, and <math>c</math> are in harmonic progression if and only if <math>1/b - 1/a = 1/c - 1/b</math>. | ||
== Properties == | == Properties == | ||
− | Because the reciprocals of the terms in a harmonic sequence are in arithmetic progression, one can apply properties of arithmetic sequences to derive a general form for harmonic sequences. Namely, for some constants <math>a</math> and <math>d</math>, the terms of any harmonic sequence can be written as <cmath>\frac{1}{a}, \textrm{ } \frac{1}{a+d}, \textrm{ } \frac{1}{a+2d}, \textrm{ } \cdots \textrm{ } \frac{1}{a+(n-1)d}.</cmath> | + | Because the reciprocals of the terms in a harmonic sequence are in arithmetic progression, one can apply properties of arithmetic sequences to derive a general form for harmonic sequences. Namely, for some constants <math>a</math> and <math>d</math>, the terms of any finite harmonic sequence can be written as <cmath>\frac{1}{a}, \textrm{ } \frac{1}{a+d}, \textrm{ } \frac{1}{a+2d},\textrm{ } \cdots \textrm{ }, \frac{1}{a+(n-1)d}.</cmath> |
− | A common lemma is that a sequence is in harmonic progression if and only if <math>a_n</math> is the harmonic mean of <math>a_{n-1}</math> and <math>a_{n+1}</math> | + | A common lemma is that a sequence is in harmonic progression if and only if <math>a_n</math> is the harmonic mean of <math>a_{n-1}</math> and <math>a_{n+1}</math> for any consecutive terms <math>a_{n-1}, a_n, a_{n+1}</math>. In symbols, <math>2/a_n = 1/a_{n-1} + 1/a_{n+1}</math>. This is mostly used to perform substitutions, though it occasionally serves as a definition of harmonic sequences. |
== Sum == | == Sum == | ||
− | A ''harmonic series'' is the sum of all the terms in a harmonic series. All infinite harmonic series diverges | + | A '''harmonic series''' is the sum of all the terms in a harmonic series. All infinite harmonic series diverges, which follows by the limit comparison test with the series <math>1 + 1/2 + 1/3 + \cdots</math>. This series is referred to as ''the'' [[harmonic series]]. As for finite harmonic series, there is no known general expression for their sum; one must find a strategy to evaluate it on a case-by-case basis. |
== Examples == | == Examples == |
Revision as of 20:56, 26 November 2021
In algebra, a harmonic sequence, sometimes called a harmonic progression, is a sequence of numbers such that the difference between the reciprocals of any two consecutive terms is constant. In other words, a harmonic sequence is formed by taking the reciprocals of every term in an arithmetic sequence.
For example, and are harmonic sequences; however, and are not.
More formally, a harmonic progression biconditionally satisfies A similar definition holds for infinite harmonic sequences. It appears most frequently in its three-term form: namely, that constants , , and are in harmonic progression if and only if .
Contents
[hide]Properties
Because the reciprocals of the terms in a harmonic sequence are in arithmetic progression, one can apply properties of arithmetic sequences to derive a general form for harmonic sequences. Namely, for some constants and , the terms of any finite harmonic sequence can be written as
A common lemma is that a sequence is in harmonic progression if and only if is the harmonic mean of and for any consecutive terms . In symbols, . This is mostly used to perform substitutions, though it occasionally serves as a definition of harmonic sequences.
Sum
A harmonic series is the sum of all the terms in a harmonic series. All infinite harmonic series diverges, which follows by the limit comparison test with the series . This series is referred to as the harmonic series. As for finite harmonic series, there is no known general expression for their sum; one must find a strategy to evaluate it on a case-by-case basis.
Examples
Here are some example solutions that utilize harmonic sequences and series.
Example 1
Find all real numbers such that is a harmonic sequence.
Solution: Using the harmonic mean properties of harmonic sequences, Note that would create a term of —something that breaks the definition of harmonic sequences—which eliminates them as possible solutions. We can thus multiply both sides by to get . Expanding these factors yields , which simplifies to . Thus, is the only solution to the equation, as desired.
Example 2
Let , , and be positive real numbers. Show that if , , and are in harmonic progression, then , , and are as well.
Solution: Using the harmonic mean property of harmonic sequences, we are given that , and we wish to show that . We work backwards from the latter equation.
One approach might be to add to both sides of the equation, which gives us Because , , and were all defined to be positive, . Thus, we can divide both sides of the equation by to get , which was given as true.
From here, it is easy to write the proof forwards. Doing so yields that , which implies that , , and is in harmonic progression, as required.
Example 3
2019 AMC 10A Problem 15: A sequence of numbers is defined recursively by , , and for all Then can be written as , where and are relatively prime positive integers. What is ?
Solution: We simplify the series recursive formula. Taking the reciprocals of both sides, we get the equality Thus, . By an above lemma, the entire sequence is in harmonic progression, which means that we can apply tools of harmonic sequences to this problem.
We will now find a closed expression for the sequence. Let and . Simplifying the first equation yields and substituting this into the second equation yields . Thus, and so . The answer is then , or .
More Problems
Here are some more problems that utilize harmonic sequences and series. Note that harmonic sequences are rather uncommon compared to their arithmetic and geometric counterparts.