Problem 35 - Thomas Calculus

Question

In Exercises \(31-38,\) use the \(n\) th-Term Test for divergence to show that the series is divergent, or state that the test is inconclusive. $$\sum_{n=1}^{\infty} \cos \frac{1}{n}$$

Step-by-Step Solution

Answer

The series diverges by the nth-Term Test, as the limit is 1.

Step 1: State the nth-Term Test for Divergence

The nth-term test for divergence states that if \( \lim_{{n \to \infty}} a_n eq 0 \), then the series \( \sum_{n=1}^{\infty} a_n \) is divergent. If \( \lim_{{n \to \infty}} a_n = 0 \), the test is inconclusive.

Step 2: Identify the terms of the series

The terms of the series are given by \( a_n = \cos \frac{1}{n} \).

Step 3: Evaluate the limit of the sequence terms

Compute \( \lim_{{n \to \infty}} \cos \frac{1}{n} \). Since \( \frac{1}{n} \to 0 \) as \( n \to \infty \), and \( \cos x \to 1 \) as \( x \to 0 \), it follows that \( \lim_{{n \to \infty}} \cos \frac{1}{n} = 1 \).

Step 4: Apply the nth-Term Test

Since \( \lim_{{n \to \infty}} \cos \frac{1}{n} = 1 eq 0 \), we can conclude that the nth-term test shows the series \( \sum_{n=1}^{\infty} \cos \frac{1}{n} \) is divergent.

Key Concepts

Divergent Series

Understanding divergent series is a key part of calculus and analysis. A series can be thought of as an infinite sum of terms. When the sum of all these terms doesn't tend to a finite limit, the series is called divergent.

This occurs when a series grows infinitely large or oscillates without approaching any fixed value.

Here's what you should know about divergent series:

• A divergent series cannot be assigned a numerical value in the usual sense.
• Even though individual terms might get smaller, the series sum keeps getting larger or oscillates.
• The nth-Term Test for Divergence is a way to identify some divergent series.

For a series to diverge, using the nth-term test, its term's limit as it goes to infinity cannot be zero. If the limit is anything other than zero, then the series is divergent.

Limit of a Sequence

A fundamental concept when dealing with series and sequences is the limit of a sequence. The limit helps determine the behavior of a sequence as the number of terms grows larger and larger.

The limit of a sequence \( a_n \), written as \( \lim_{{n \to \infty}} a_n \), describes what value \( a_n \) gets closer and closer to, as \( n \) becomes infinitely large.

Here’s how you can understand limits further:

• If \( \lim_{{n \to \infty}} a_n = L \), then eventually terms in the sequence get arbitrarily close to \( L \).
• If \( \lim_{{n \to \infty}} a_n eq 0 \), many tests like the nth-Term Test conclude the series diverges.
• Limits help us decide if series converge (approach a sum) or diverge.

In our example, we found \( \lim_{{n \to \infty}} \cos \frac{1}{n} = 1 \), indicating divergence, as the term's limit isn't zero.

Trigonometric Functions

Trigonometric functions, like cosine, often appear in series problems. Understanding their behaviors is vital in solving related mathematical problems.

For the cosine function, specifically:

• It oscillates between -1 and 1.
• As \( x \to 0 \), \( \cos x \to 1 \).
• This property helps determine the limits involving the cosine function.

In the context of the series \( \sum_{n=1}^{\infty} \cos \frac{1}{n} \):

The term \( \cos \frac{1}{n} \) approaches 1 as \( n \to \infty \). Thus, each term of the series doesn't approach 0, crucial for determining divergence through the nth-Term Test.

Understanding trigonometric function properties can vastly improve your comprehension of how they interact in sequences and series.