Problem 18 - Essential Calculus Early Transcendentals

Question

Determine whether the series is convergent or divergent. If it is convergent, find its sum. $$\sum_{n=1}^{\infty} \cos \frac{1}{n}$$

Step-by-Step Solution

Answer

The series diverges because the terms do not approach zero.

Step 1: Examine the General Term

Consider the general term of the series: \( \cos\left( \frac{1}{n} \right). \) As \( n \to \infty, \frac{1}{n} \to 0. \) Thus, \( \cos\left( \frac{1}{n} \right) \to \cos(0) = 1. \) Since the general term does not approach zero, this suggests potential divergence.

Step 2: Apply the Divergence Test

For a series \( \sum a_n \) to converge, the general term \( a_n \) must approach zero as \( n \to \infty. \) However, since: \[ \lim_{n \to \infty} \cos\left( \frac{1}{n} \right) = 1 eq 0, \] the series \( \sum_{n=1}^{\infty} \cos\left( \frac{1}{n} \right) \) must diverge.

Key Concepts

Divergence Test

The Divergence Test is a simple tool used to determine if an infinite series diverges or converges. It checks the behavior of the series' general term as the number of terms goes to infinity. The idea is quite straightforward. For a series \( \sum a_n \) to be convergent, the term \( a_n \) should approach zero as \( n \to \infty \).
However, if the limit does not equal zero, the series will diverge. This test is just a preliminary check. It helps us quickly rule out certain series without a lengthy calculation.
For our given series, \( \cos\left( \frac{1}{n} \right) \) tends towards 1 as \( n \to \infty \). Since this limit is not zero, the series is divergent, confirming our findings using the divergence test.

Infinite Series

An infinite series is a sum of infinitely many terms, expressed in the form \( \sum_{n=1}^{\infty} a_n \). It's similar to an arithmetic or geometric series but without a finite number of terms. When we talk about whether a series is convergent or divergent, we're considering if this sum reaches a finite limit.
There are various tests, like the divergence test, ratio test, root test, and more, to analyze convergence. In the case of the series \( \sum_{n=1}^{\infty} \cos\left( \frac{1}{n} \right) \), as discussed earlier, the terms do not tend toward zero, indicating divergence.
Finding if an infinite series has a finite sum is crucial in mathematical analysis and applications in areas such as physics and engineering.

Trigonometric Convergence

Trigonometric convergence deals with infinite series where terms involve trigonometric functions such as sine, cosine, or tangent. These functions often introduce unique behaviors due to their periodic nature.
When analyzing such series, it's significant to determine how quickly terms approach zero as the series progresses. For a trigonometric series like \( \cos\left( \frac{1}{n} \right) \), we should check if the terms decrease to zero rapidly enough.
In our example, even though \( \frac{1}{n} \) tends towards zero, the cosine function does not yield terms that trend to zero, but actually approaches one.

• This illustrates that despite the trigonometric function, convergence can’t be assumed and must be checked rigorously.
• Generally, unless extra decay is introduced (for example, by multiplying \( \cos\left( \frac{1}{n} \right) \) by \( 1/n^2 \) or similar), the series might diverge.