Problem 27 - Thomas Calculus

Question

In Exercises \(27-34,\) use the \(n\) th-Term Test for divergence to show that the series is divergent, or state that the test is inconclusive. $$ \sum_{n=1}^{\infty} \frac{n}{n+10} $$

Step-by-Step Solution

Answer

The series diverges by the nth-term test for divergence.

Step 1: Understand the nth-Term Test for Divergence

The nth-term test for divergence states that if the limit of the nth term of a series, \(a_n\), as \(n\) approaches infinity is not zero, then the series \(\sum a_n\) diverges. However, if \(\lim_{{n \to \infty}} a_n = 0\), the test is inconclusive as the series might still diverge.

Step 2: Define the nth Term

For the series \(\sum_{n=1}^{\infty} \frac{n}{n+10}\), let's define the nth term \(a_n\) as \(\frac{n}{n+10}\). We will examine the behavior of this term as \(n\) approaches infinity.

Step 3: Calculate the Limit of the nth Term

To apply the nth-term test, we need to calculate \(\lim_{{n \to \infty}} \frac{n}{n+10}\). First, divide the numerator and the denominator by \(n\): \[\lim_{{n \to \infty}} \frac{n}{n+10} = \lim_{{n \to \infty}} \frac{1}{1+\frac{10}{n}}.\]

Step 4: Evaluate the Limit

As \(n\) approaches infinity, the term \(\frac{10}{n}\) becomes 0. So, the limit simplifies to \(\frac{1}{1+0} = 1\). Thus, \(\lim_{{n \to \infty}} \frac{n}{n+10} = 1\).

Step 5: Apply the nth-Term Test Conclusion

Since \(\lim_{{n \to \infty}} \frac{n}{n+10} = 1\), which is not zero, the series \(\sum_{n=1}^{\infty} \frac{n}{n+10}\) diverges by the nth-term test.

Key Concepts

Divergent Series

A divergent series is a series that does not have a finite limit. This means that as you add more and more terms of the series together, the total sum will either grow indefinitely or fluctuate without settling on a specific value. For example, with the series \(\sum_{n=1}^{\infty} \frac{n}{n+10}\), performing the nth-term test for divergence reveals that the individual terms approach 1 as \(n\) progresses. - Since the nth term does not approach zero, the entire series cannot converge to a finite number. - The sum of all terms, in this case, continues to increase, signifying that the series is divergent.Applying the nth-term test gives us a quick way to check if a series might diverge before getting into more complex tests.

Limit of a Sequence

Understanding the limit of a sequence is essential for determining the behavior of an infinite series. The limit of a sequence when discussed in terms of series refers to what the terms of the sequence approach as the index \(n\) goes to infinity.- For the sequence \(a_n = \frac{n}{n+10}\), as \(n\) increases, the terms get closer and closer to 1.- Calculating \(\lim_{{n \to \infty}} \frac{n}{n+10}\) involves simplifying \(\frac{1}{1+\frac{10}{n}}\) and observing the behavior as \(n\) grows very large.- When \(n\) tends to infinity, the \(\frac{10}{n}\) term approaches zero, resulting in a limit of 1. This limit confirms that the sequence's terms do not shrink to zero, a key indicator required by the nth-term test to suggest divergence.

Infinite Series

An infinite series is essentially an ordered list of numbers that extends indefinitely. Adding up all these terms, we see whether this sum results in a finite number or not.- For instance, the infinite series in question, \(\sum_{n=1}^{\infty} \frac{n}{n+10}\), is formed by repeatedly adding terms like \(\frac{1}{11}\), \(\frac{2}{12}\), and so forth.- Infinite series can either converge or diverge. If the total converges, it means it settles to a specific value even with infinite terms. - From the nth-term test, if any term in an infinite series fails to approach zero, then the series absolutely cannot converge, which leads us to conclude that it must diverge.Understanding how and why such a series behaves in infinite sums is crucial to grasp broader mathematical concepts and applications.