Definition:Alternating Series

Definition

Let $\ds s = \sum_{n \mathop = 1}^\infty a_n$ be a series in the real numbers $\R$.

The series $s$ is an alternating series if and only if the terms of $\sequence {a_n}$ alternate between positive and negative.

Examples

General Form

Let $S_n$ be a real series of the form:

$\ds S_n = \sum_{k \mathop \ge 0} \paren {-1}^k p_k$

or:

$\ds S_n = \sum_{k \mathop \ge 1} \paren {-1}^{k - 1} p_k$

where:

$\forall k \in \N: p_k > 0$

Then $S_n$ is an alternating series.

Also see

• Results about alternating series can be found here.

Sources

• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): alternating series
• 1992: Larry C. Andrews: Special Functions of Mathematics for Engineers (2nd ed.) ... (previous) ... (next): $\S 1.2.2$: Summary of convergence tests
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): alternating series
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): alternating series
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): alternating series
• 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): alternating series