Absolute and conditional convergence

 Objective: Determine whether a series is absolutely convergent or conditionally 

Theorem

Example. Determine whether the following series converges absolutely, conditionally or diverges.

Solution

Method

1) We start by calculating the series of the absolute value of the general term of the given series

2) We find a new series of which we study the convergence or divergence

3) If this series is convergent, the series of the absolute value of the general term is convergent. Therefore the given series converges absolutely

4) If this series diverges as in the example above, the series of the absolute value of the general term diverges. The series doesn't converge absolutely.  In this case we study the convergence of the given series. We conclude that the series converges conditionally.

Practice

Determine if the following series converges absolutely, conditionally or diverges.

The alternating series test

 Objectives:

1) Define the alternating series

2) Use the alternating series test for convergence

Introduction

Any series whose terms alternate in sign is called an alternating series.

Examples:

Definition

An alternate series is a series whose terms alternate between positive and negative values. An alternating series can be written in either form where bₙ is positive for all positive integers n

Theorem

Examples

Solution

a. Let's apply the condition above. First, let's see if the first condition is verified.

We have : bₙ = 1/n². Therefore bₙ₊₁ = 1/(n+1)² = 1/n² + 2n +1 >0 for all n≥1 and bₙ.₊₁≤ bₙ The first condition is verified. Let's see if the second condition is verified:

b.  In this problem we have bₙ₊₁>bₙ. We can prove that by calculating bₙ₊₁ -bₙ. The first condition not being verified, therefore we cannot apply the alternating series test. Instead let's find the limit of the sequence of the partial sum or the limit of the general term of the partial sum. We have:

 

The series is then divergent.

Practice