Definition:Harmonic Sequence

This page is about Harmonic Sequence. For other uses, see Harmonic.

Definition

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:

$\dfrac 1 a, \dfrac 1 {a + d}, \dfrac 1 {a + 2 d}, \dfrac 1 {a + 3 d}, \ldots$

Initial Term

The term $a$ is the initial term of $\sequence {a_k}$.

Common Difference

The term $d$ is the common difference of $\sequence {a_k}$.

Also see

• Results about harmonic sequences can be found here.

Sources

• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): harmonic progression
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): harmonic sequence (harmonic progression)
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): harmonic sequence (harmonic progression)
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): harmonic sequence
• 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): harmonic sequence (harmonic progression)

• Weisstein, Eric W. "Harmonic Series." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/HarmonicSeries.html