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Alternating Series 📂Calculus

Alternating Series

Definition

A series in which the sign of each term alternates is called an alternating series. In other words, for $b_{n} \gt 0$, a series whose general term is expressed in the following form.

$$ a_{n} = (-1)^{n-1}b_{n} \qquad \text{ or } \qquad a_{n} = (-1)^{n}b_{n} $$

Explanation

One method to determine the convergence of an alternating series is the Alternating Series Test.

Alternating Series Test

An alternating series $\sum\limits_{n = 1}^{\infty} (-1)^{n-1}b_{n}$ $(b_{n} \gt 0)$ that satisfies the following conditions converges.

  1. $b_{n+1} \le b_{n} \quad \forall n$.
  2. $\lim\limits_{n \to \infty} b_{n} = 0$.

Examples

Alternating Harmonic Series

The alternating harmonic series converges.

$$ \sum\limits_{n = 1}^{\infty} (-1)^{n-1}\dfrac{1}{n} = \ln 2 $$