Alternating Series Test for Convergence

Benjamin Wright

4 min read

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Study Guide Overview

This study guide covers the Alternating Series Test for convergence in AP Calculus BC. It explains the theorem, which states conditions for convergence (limit of a_n = 0 and a_n decreases). It provides a breakdown of the theorem with the alternating harmonic sequence as an example and includes practice problems and solutions for applying the alternating series test.

#10.7 Alternating Series Test for Convergence

Welcome to AP Calc 10.7! In this lesson, you’ll how to test for convergence when dealing with an alternating series.

#➕ Alternating Series Test Theorem

The alternating series test for convergence states that for an alternating series $\sum(-1)^n\cdot a_n$ , if

$1. \lim_{n\to \infty} a_n=0 \ \text{and}$

$2. \ a_n \ \text{decreases,}$

then the series converges. Otherwise, it diverges.

#🧱 Breaking Down the Theorem

To illustrate this theorem, let’...

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Question 1 of 10

What are the two conditions that must be met for an alternating series $\sum (-1)^n a_n$ to converge according to the Alternating Series Test? 🤔

$\lim_{n\to \infty} a_n = 1$ and 2. $a_n$ is increasing

$\lim_{n\to \infty} a_n = 0$ and 2. $a_n$ is decreasing

$\lim_{n\to \infty} a_n = \infty$ and 2. $a_n$ is decreasing

$\lim_{n\to \infty} a_n = 0$ and 2. $a_n$ is increasing

Question 1 of 10

What are the two conditions that must be met for an alternating series $\sum (-1)^n a_n$ to converge according to the Alternating Series Test? 🤔

$\lim_{n\to \infty} a_n = 1$ and 2. $a_n$ is increasing

$\lim_{n\to \infty} a_n = 0$ and 2. $a_n$ is decreasing

$\lim_{n\to \infty} a_n = \infty$ and 2. $a_n$ is decreasing

$\lim_{n\to \infty} a_n = 0$ and 2. $a_n$ is increasing