Problem 25 - Thomas Calculus

Question

Which of the series in Exercises 13 46 converge, and which diverge? Give reasons for your answers. (When you check an answer, remember that there may be more than one way to determine the series' convergence or divergence.) $$ \sum_{n=0}^{\infty} \frac{-2}{n+1} $$

Step-by-Step Solution

Answer

The series diverges, as it resembles the harmonic series (which diverges) multiplied by a constant.

Step 1: Identify the series

The series given is: \( \sum_{n=0}^{\infty} \frac{-2}{n+1} \). It is a straightforward infinite series where the terms are of the form \( a_n = \frac{-2}{n+1} \).

Step 2: Check the conditions for divergence

A series \( \sum a_n \) diverges if the general term \( a_n \) does not approach zero as \( n \to \infty \). Calculate \( \lim_{n \to \infty} \frac{-2}{n+1} \).

Step 3: Calculate the limit

Compute the limit: \( \lim_{n \to \infty} \frac{-2}{n+1} = 0 \). Though these terms approach 0, we need to apply additional tests because this alone does not confirm convergence.

Step 4: Use Divergence Test

For a series applying the divergence test, if the limit of \( a_n \) does not equal 0, then the series must diverge. Here, \( lim_{n \to \infty} \frac{-2}{n+1} = 0 \), so we need another test.

Step 5: Recognize the Harmonic Series and its Divergence

Observe that the series \( \sum \frac{-2}{n+1} = -2 \sum \frac{1}{n+1} \) resembles the harmonic series. The harmonic series \( \sum \frac{1}{n} \) is known to diverge. Thus, multiplying it by a constant (\(-2\)) will still result in divergence.

Key Concepts

Infinite Series

An infinite series is the sum of infinitely many terms that follow a specific pattern. Think of it like a never-ending sequence of numbers added together. In mathematical terms, it is represented as:

• \( \sum_{n=0}^{\infty} a_n \)

Each term in the series can change depending on the index \( n \), which typically starts from 0 and goes to infinity. Infinite series can either converge or diverge. This means they can either reach a specific number (converge) or keep increasing or decreasing indefinitely (diverge). Infinite series are foundational in calculus and help in understanding various mathematical functions and solutions.

Harmonic Series

The harmonic series is a famous example of an infinite series defined as:

• \( \sum_{n=1}^{\infty} \frac{1}{n} \)

Each term of this series gets smaller as \( n \) increases. You might assume that since the terms decrease, the series might sum to a fixed number. However, surprisingly, the harmonic series diverges. When you add its terms, they will grow larger without bound, even though they increase very slowly. The concept is also applied in exercises where you recognize similar series patterns that diverge due to their resemblance to this fundamental series.

Divergence Test

To determine if an infinite series diverges, one common method is the divergence test. The idea is straightforward: if the limit of the terms \( a_n \) as \( n \to \infty \) is not zero, then the series definitely diverges. However, the opposite isn't necessarily true:

• If \( \lim_{n \to \infty} a_n = 0 \), the test is inconclusive.
• In this case, it might still converge or diverge.

This test is often the first step in checking the behavior of an infinite series. If a series fails the divergence test by having non-zero limit terms, it clearly shows divergence.

Limit of a Series

The limit of a series is an essential concept in determining whether a series converges or diverges. We often look at the limit of the general term \( a_n \) as \( n \to \infty \):

• \( \lim_{n \to \infty} a_n \)

If this limit is not zero, the series diverges. However, when the limit is zero, further testing is needed to reach a concrete conclusion about converging or diverging. In some cases, although \( a_n \to 0 \), the series still diverges, as seen with the harmonic series. Calculating limits helps us gain understanding of the behavior and potential sum of an infinite series.