Problem 29 - Essential Calculus Early Transcendentals

Question

Determine whether the series is convergent or divergent. $$\sum_{n=1}^{\infty} \sin \left(\frac{1}{n}\right)$$

Step-by-Step Solution

Answer

The series \( \sum_{n=1}^{\infty} \sin \left(\frac{1}{n}\right) \) is divergent.

Step 1: Define Convergence and Divergence

A series \( \sum a_n \) is convergent if its sequence of partial sums \( S_n = a_1 + a_2 + ... + a_n \) approaches a definite number as \( n \rightarrow \infty \). If the sequence of partial sums does not approach a definite number, the series is divergent.

Step 2: Analyze Series Term

Consider the general term of the series \( a_n = \sin \left(\frac{1}{n}\right) \). As \( n \to \infty \), the argument \( \frac{1}{n} \to 0 \) and thus \( \sin \left(\frac{1}{n}\right) \to \sin(0) = 0 \). However, this does not determine if the series converges.

Step 3: Find Equivalent Form

Since \( \sin(x) \approx x \) when \( x \) is near zero, \( \sin \left(\frac{1}{n}\right) \approx \frac{1}{n} \) for large \( n \). Hence, \( a_n = \sin \left(\frac{1}{n}\right) \sim \frac{1}{n} \).

Step 4: Compare With Harmonic Series

The harmonic series \( \sum \frac{1}{n} \) is a well-known divergent series. Since \( a_n \approx \frac{1}{n} \), the series \( \sum \sin \left(\frac{1}{n}\right) \) behaves similarly to the harmonic series.

Step 5: Apply Limit Comparison Test

Using the limit comparison test: if \( \lim_{{n \to \infty}} \frac{a_n}{b_n} = c \) where \( 0 < c < \infty \), then both series \( \sum a_n \) and \( \sum b_n \) either converge or diverge together. Here, \( \frac{a_n}{b_n} = \frac{\sin \left(\frac{1}{n}\right)}{\frac{1}{n}} = n \sin \left(\frac{1}{n}\right) \rightarrow 1 \) as \( n \to \infty \). Since \( \sum \frac{1}{n} \) diverges, \( \sum \sin \left(\frac{1}{n}\right) \) also diverges.

Key Concepts

Limit Comparison Test

The Limit Comparison Test is a valuable tool for determining the convergence or divergence of an infinite series. It works best when you can compare the series in question to another series whose behavior is well known. To use this test, you analyze the limit of the ratio of the terms of the two series as they go to infinity.

For example, suppose you have two series, \( \sum a_n \) and \( \sum b_n \), and you want to know if \( \sum a_n \) converges or diverges. With the Limit Comparison Test, you compare term by term:

• Compute \( \lim_{n \to \infty} \frac{a_n}{b_n} \).
• If this limit \( c \) exists and is a finite constant \( 0 < c < \infty \), then both series \( \sum a_n \) and \( \sum b_n \) converge or diverge together.

This method simplifies situations where the series doesn't immediately reveal its convergent or divergent nature. In our exercise, it was crucial to compare \( \sin \left(\frac{1}{n}\right) \) to the harmonic series \( \frac{1}{n} \) to conclude divergence.

Harmonic Series

The harmonic series is a simple yet profound series that provides a classic example of divergence in mathematics. It is given by:\[\sum_{n=1}^{\infty} \frac{1}{n}\]This series lacks a finite sum, meaning as you add more terms, the total keeps increasing without bound. Thus, we say the harmonic series diverges.

The importance of the harmonic series lies in how often it's used as a benchmark for comparing other series. **Why is it relevant?**

• **Known Divergence:** Its divergent nature is understood and proven through simple integral tests.
• **Behavior Near Zero:** The terms \( \frac{1}{n} \) approach 0 as \( n \to \infty \), yet the series diverges.

Understanding the harmonic series helps in analyzing any new series that seems similar, as seen with \( \sin \left(\frac{1}{n}\right) \) which is comparable when \( n \) is large.

Partial Sums

When we're discussing series, we often talk about partial sums to understand whether a series converges or diverges. A partial sum is essentially a finite sum of the first few terms of an infinite series, denoted as:\[S_n = a_1 + a_2 + \ldots + a_n\]The concept of partial sums is fundamental because it allows us to look closer at how the series behaves as you add more terms.

**What to Note:**

• **Convergence:** If as \( n \to \infty \), the sequence of partial sums \( \{S_n\} \) approaches a definite number, the series converges.
• **Divergence:** Conversely, if \( \{S_n\} \) does not approach a definite number, the series is divergent.

In our exercise, although each term \( \sin \left(\frac{1}{n}\right) \) approaches 0, the series diverges because its partial sums \( \sum_{n=1}^{N} \sin \left(\frac{1}{n}\right) \) grow without bound, resembling the partial sums of a harmonic series.