Strategies for testing the convergence or divergence of an infinite series

 Studying infinite series comes down to the study of their convergence or divergence. Unfortunately in this section of mathematics there is no definite method to determine the convergence or divergence of a series. In this post I show the strategies used for that purpose.

Many series fall into several types. Recognizing these types will help to know which test or strategies to use to determine the convergence or divergence a series.

Some series like the p-series and the geometric series are easy to determine their convergence.   

P-series

Comparison and limit comparison test

If a series is similar to either one of the series above, you can use the comparison or the limit comparison test.

Some series will not converge obviously. You will save a lot of work by recognizing this.

Divergence test 

The following two tests prove convergence but also the stronger fact that Σ❘aₙ| converges but also converges absolutely.

The above example contains powers and factorials. The ratio test is useful when aₙ contains powers and factorials

Using convergence steps. Examples

Example 1

Solution

We take the following steps:

1. The series is not an harmonic series, an alternating harmonic series a geometric or p-series

2. It is not an alternating series

3. It is not similar to a p-series or a geometric but we can approximate it to an harmonic series:

For larger n the series can be approximated as follows:

The series is similar to an harmonic series. Therefore we can use the comparison test or the limit comparison test. Using the limit comparison test we have:

The limit is a finite number. Since the harmonic series diverges, the given series diverges also.

Example 2

Solution

We take the following steps:

1. The series is not a familiar series: harmonic, p-series, geometric series

2. It is not an alternating series

3. There is no known series to compare it to

4. There is no factorial. 

5. There is a power but we cannot use the root test

6. Having used all the options for convergence or divergence let's use the divergence test

We have:

The series diverges according to the divergence test.

Practice

Determine whether the following series converges or diverges using the appropriate steps.

The Root Test

 Objectives:

1) Define the root test

2) Use the root test to determine if a series converges or diverges

Definition

The ratio test uses the limit of the square root of the absolute value of a series to determine its convergence or divergence. It is used in series involving exponentials.

Theorem 

Examples

Solutions

a. Let's determine ρ:

Since ρ<1, the series converges absolutely.

b. Let's determine p:

ρ = ∞/∞ (indetermination)

Let's get rid of the indetermination by applying the l'Hospital rule:

The series diverges.

Practice