Problem 18 - Calculus Single Variable

Question

State what conclusion, if any, may be drawn from the Divergence Test. $$ \sum_{n=1}^{\infty} \cos (1 / n) $$

Step-by-Step Solution

Answer

The series diverges by the Divergence Test as the limit of the terms is not zero.

Step 1: Write the Terms of the Series

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

Step 2: Assess the Limit as n Approaches Infinity

Consider the limit \(\lim_{{n \to \infty}} \cos\left(\frac{1}{n}\right)\). As \(n\) becomes very large, \(\frac{1}{n}\) approaches zero, and thus, \(\cos\left(\frac{1}{n}\right)\) approaches \(\cos(0) = 1\).

Step 3: Apply the Divergence Test

The Divergence Test states that if \(\lim_{{n \to \infty}} a_n eq 0\), the series \(\sum_{n=1}^{\infty} a_n\) diverges. Here, since the limit is 1 (not zero), the test implies that the series \(\sum_{n=1}^{\infty} \cos \left(\frac{1}{n}\right)\) diverges.

Key Concepts

Infinite Series

An infinite series is the sum of an endless sequence of numbers, where each term in the series is defined by a particular function. Typically, an infinite series is represented as \( \sum_{n=1}^{\infty} a_n \), where \( a_n \) is the nth term of the sequence. The concept is deeply rooted in calculus and analysis, as it helps in understanding sequences that extend indefinitely.
When dealing with infinite series, the primary goal is determining whether the series converges or diverges. Convergence in this context means that as we add more and more terms, the overall sum approaches a fixed value. Divergence indicates that the sum does not settle at any particular number.
In the provided exercise, the series is expressed as \( \sum_{n=1}^{\infty} \cos \left(\frac{1}{n}\right) \). Each term is defined by the function \( a_n = \cos\left(\frac{1}{n}\right) \). As we progress through the sequence, we inspect how these numbers add up over infinite terms and whether a sum can be precisely nailed down or not.

Convergence

Convergence is a crucial notion when assessing infinite series. It describes a situation where the series grows closer to a specific value as more terms are added. Mathematically, this means the partial sums of the series approach a finite limit.
A series converges if its sequence of partial sums, \( S_n = a_1 + a_2 + ... + a_n \), approaches a limit as \( n \) becomes very large. If such a limit exists, and it's finite, the series is convergent.

• Conversely, if no such limit is present, the series diverges.

When applying the Divergence Test, one primary focus is this aspect of convergence. For the series \( \sum_{n=1}^{\infty} \cos \left(\frac{1}{n}\right) \), since we found the limit of the terms \( a_n \) to be 1 (not zero), we can conclude that the series does not converge. It doesn't hone in on a finite limit as \( n \) approaches infinity.

Limit of a Sequence

Understanding the limit of a sequence plays a pivotal role in discerning the behavior of an infinite series. A sequence can be thought of as a listed collection of numbers in a specified order. The limit of a sequence, \( \lim_{{n \to \infty}} a_n \), is the value that the elements of the sequence approach as \( n \) becomes exceedingly large.
For the series discussed in the exercise, each term is given by \( a_n = \cos\left(\frac{1}{n}\right) \). As \( n \) grows, \( \frac{1}{n} \) naturally tends to zero. Consequently, \( \cos(\frac{1}{n}) \) approaches \( \cos(0) = 1 \). This step is crucial because the Divergence Test relies on examining the term limits.

• If the limit of the sequence as \( n \to \infty \) is not zero, then it directly informs us that the infinite series it's associated with must diverge.

Therefore, knowing that the limit here is 1, we affirm that the series \( \sum_{n=1}^{\infty} \cos\left(\frac{1}{n}\right) \) cannot possibly converge, based on the test's guidelines.