Definition:Harmonic Progression
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This page is about Harmonic Progression. For other uses, see Harmonic.
Definition
The term harmonic progression is used to mean one of the following:
Harmonic Sequence
A harmonic sequence is a sequence $\sequence {a_k}$ in $\R$ defined as:
- $h_k = \dfrac 1 {a + k d}$
where:
- $k \in \set {0, 1, 2, \ldots}$
- $-\dfrac a d \notin \set {0, 1, 2, \ldots}$
Thus its general form is:
- $\dfrac 1 a, \dfrac 1 {a + d}, \dfrac 1 {a + 2 d}, \dfrac 1 {a + 3 d}, \ldots$
Harmonic Series
The series defined as:
- $\ds \sum_{n \mathop = 1}^\infty \frac 1 n = 1 + \frac 1 2 + \frac 1 3 + \frac 1 4 + \cdots$
is known as the harmonic series.
Sources
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): harmonic progression