Definition:Arithmetic Series

Definition

An arithmetic series is a series whose underlying sequence is an arithmetic sequence:

\(\ds S_n\)

\(=\)

\(\ds \sum_{k \mathop = 0}^{n - 1} a + k d\)

\(\ds \)

\(=\)

\(\ds a + \paren {a + d} + \paren {a + 2 d} + \cdots + \paren {a + \paren {n - 1} d}\)

Also see

• Sum of Arithmetic Sequence

• Results about arithmetic series can be found here.

Linguistic Note

In the context of an arithmetic sequence or arithmetic-geometric sequence, the word arithmetic is pronounced with the stress on the first and third syllables: a-rith-me-tic, rather than on the second syllable: a-rith-me-tic.

This is because the word is being used in its adjectival form.

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 19$: Arithmetic Series: $19.1$
• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): arithmetic series
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): arithmetic series
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): arithmetic series
• 2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 21$: Series of Constants: Arithmetic Series: $21.1.$
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): arithmetic series
• 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): arithmetic series