Problem 16 - 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.) $$ \sum_{n=1}^{\infty} \frac{5}{n+1} $$

Step-by-Step Solution

Answer

The series diverges.

Step 1: Identify the Series

The series given is \( \sum_{n=1}^{\infty} \frac{5}{n+1} \). This is a series in which each term is \( \frac{5}{n+1} \).

Step 2: Use Comparison Test

The series can be compared to the harmonic series \( \sum_{n=1}^{\infty} \frac{1}{n} \), which is known to diverge. Since \( \frac{5}{n+1} \) behaves similarly to \( \frac{5}{n} \) for large \( n \), the comparison test can be applied here.

Step 3: Simplify the Comparison

Notice that \( \frac{5}{n+1} > \frac{5}{n+2} \) for \( n \geq 1 \), and \( \frac{5}{n+2} \) diverges as it's similar to \( \frac{5}{n} \). Since a divergent series is smaller, \( \frac{5}{n+1} \) likely diverges.

Step 4: Conclude the Divergence

Since \( \sum_{n=1}^{\infty} \frac{1}{n} \) diverges and \( \frac{5}{n+1} \geq \frac{1}{n+2} \) for all \( n \geq 1 \), by the comparison test, \( \sum_{n=1}^{\infty} \frac{5}{n+1} \) also diverges.

Key Concepts

Comparison Test

The comparison test is a method used to determine the convergence or divergence of a series. It is particularly useful when you can compare a complex series with a simpler one, whose behavior is already known. There are two forms to this test: the Direct Comparison Test and the Limit Comparison Test.

• **Direct Comparison Test**: This involves directly comparing the terms of your series with another series that is already known to converge or diverge. If a series \( \sum a_n \) has terms that are greater than or equal to the terms of a known divergent series \( \sum b_n \), then \( \sum a_n \) also diverges.


• **Limit Comparison Test**: This test uses the limit of the ratio of terms from your series and a known series. If \( c = \lim_{{n \to \infty}} \frac{a_n}{b_n} \) is a positive finite number, then both series either converge or diverge together.

In the original exercise, the series \( \sum_{n=1}^{\infty} \frac{5}{n+1} \) was compared with the harmonic series \( \sum_{n=1}^{\infty} \frac{1}{n} \). Since the harmonic series is known to diverge and for large \( n \), \( \frac{5}{n+1} \approx \frac{5}{n} \), the comparison test implies that the original series diverges as well.

Harmonic Series

The harmonic series is one of the simplest examples of a divergent series. It is represented as \( \sum_{n=1}^{\infty} \frac{1}{n} \). Despite the terms getting closer and closer to zero as \( n \) increases, the sum of the series grows without bound, indicating divergence.

Key properties of the harmonic series include:

• Each term \( \frac{1}{n} \) is the reciprocal of an integer \( n \).


• The series grows logarithmically, meaning its partial sums increase slowly, yet they continue to approach infinity.


• It is a classic example illustrating that a series can have terms that tend to zero and still diverge.

In comparing the series \( \sum_{n=1}^{\infty} \frac{5}{n+1} \) to the harmonic series, notice that each term \( \frac{5}{n+1} \) is similar in structure to \( \frac{1}{n} \), providing insight into the original series' behavior.

Divergent Series

A series is said to be divergent if the sequence of its partial sums does not converge to a finite limit. In simpler terms, as you add more and more terms of the series together, the total sum either grows without bounds or oscillates without settling at a particular value.

• One common method to test for divergence is by comparing with known divergent series like the harmonic series.


• If the terms of a series do not tend to zero, it immediately indicates divergence as a necessary condition for convergence is not met.


• Divergent does not imply there is no pattern; rather, the sum doesn't stabilize at a particular value.

In the exercise at hand, the series \( \sum_{n=1}^{\infty} \frac{5}{n+1} \) diverges, as it shares properties with the harmonic series. Specifically, since it is greater than or similar in behavior to a known divergent series, it must also diverge by the comparison test.