AP Calculus BC Study Guide & Notes

Introduction

Welcome to this guide on Absolute Convergence and Conditional Convergence, this article will guide you through the following:

What is Absolute Convergence?
What is Conditional Convergence?
How to determine if a series is absolute convergence or conditional convergence?

Let $\sum a_{n}$ be an infinite series, if

$\sum ∣ a_{n} ∣$ and $\sum a_{n}$ both converge, then the series is absolutely convergent.
$\sum a_{n}$ converges but $\sum ∣ a_{n} ∣$ diverge, then the series is conditionally convergent.

Example 1:

Determine if the series converges absolutely, conditionally, or diverge

$n = 1 \sum \infty \frac{( - 1 ) ^{n}}{3 n}$

Apply the Alternating series test to this series

Which means by Alternating Series Test, this series converges.

By p-series test, $n = 1 \sum \infty \frac{1}{3 n}$ diverge, which means the original series converges conditionally.

Example 2:

Determine if the series converge absolutely, conditionally or diverge

$n = 1 \sum \infty \frac{sin n}{n ^{2}}$

Note that this series is not an alternating series. By direct comparison test, we have:

$n = 1 \sum \infty \frac{sin n}{n ^{2}} \leq n = 1 \sum \infty \frac{1}{n ^{2}}$

Meaning the original series converges absolutely

Determine if the following is absolute convergent or conditional convergent