logo

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 bn>0b_{n} \gt 0, a series whose general term is expressed in the following form.

an=(−1)n−1bn or an=(−1)nbn 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 ∑n=1∞(−1)n−1bn\sum\limits_{n = 1}^{\infty} (-1)^{n-1}b_{n} (bn>0)(b_{n} \gt 0) that satisfies the following conditions converges.

  1. bn+1≤bn∀nb_{n+1} \le b_{n} \quad \forall n.
  2. lim⁡n→∞bn=0\lim\limits_{n \to \infty} b_{n} = 0.

Examples

Alternating Harmonic Series

The alternating harmonic series converges.

∑n=1∞(−1)n−11n=ln⁡2 \sum\limits_{n = 1}^{\infty} (-1)^{n-1}\dfrac{1}{n} = \ln 2