Problem 31 - Calculus: Early Transcendentals

Question

Determine whether the series converges or diverges. \( \displaystyle \sum_{n = 1}^{\infty} \sin \left( \frac {1}{n} \right) \)

Step-by-Step Solution

Answer

The series diverges.

Step 1: Understanding the Series

The given series is \( \displaystyle \sum_{n = 1}^{\infty} \sin \left( \frac {1}{n} \right) \). This is an infinite series of sinusoidal terms. Our task is to determine whether this series converges or diverges.

Step 2: Analyzing the Behavior of \( \sin \left( \frac{1}{n} \right) \)

We know that \( \sin(x) \approx x \) for very small values of \( x \). Hence, \( \sin \left( \frac{1}{n} \right) \approx \frac{1}{n} \) when \( n \) is large.

Step 3: Compare with the Harmonic Series

The harmonic series \( \sum_{n=1}^{\infty} \frac{1}{n} \) is a well-known divergent series. Since \( \sin \left( \frac{1}{n} \right) \approx \frac{1}{n} \), we can infer that the series \( \sum \sin \left( \frac{1}{n} \right) \) behaves similarly to the harmonic series.

Step 4: Applying the Limit Comparison Test

The Limit Comparison Test can be used to compare the series \( \sum \sin \left( \frac{1}{n} \right) \) to the harmonic series \( \sum \frac{1}{n} \). For \( a_n = \sin \left( \frac{1}{n} \right) \) and \( b_n = \frac{1}{n} \), we find \[ \lim_{n \to \infty} \frac{a_n}{b_n} = \lim_{n \to \infty} \frac{\sin \left( \frac{1}{n} \right)}{\frac{1}{n}} = \lim_{n \to \infty} \frac{\sin \left( \frac{1}{n} \right)}{\frac{1}{n}} = 1. \] Thus, since this limit is a positive finite number, both series converge or diverge together.

Step 5: Conclusion: Divergence

Since the harmonic series \( \sum \frac{1}{n} \) diverges and the limit comparison test confirmed similarity in behavior, the series \( \sum_{n = 1}^{\infty} \sin \left( \frac {1}{n} \right) \) also diverges.

Key Concepts

Limit Comparison Test

The Limit Comparison Test is a fundamental tool used to determine the convergence or divergence of an infinite series. This test involves comparing a given series to a second series whose convergence properties are already known. For two series with terms \(a_n\) and \(b_n\), where the limit \(\lim_{n \to \infty} \frac{a_n}{b_n}\) is calculated:

• If the limit is a positive finite number, both series will either converge or diverge together.
• If the limit is zero and \(b_n\) converges, then \(a_n\) also converges.
• If the limit is infinite and \(b_n\) diverges, then \(a_n\) also diverges.

In our exercise, we compared \(\sin \left( \frac{1}{n} \right)\) with \(\frac{1}{n}\), using their known behaviors to draw a conclusion.

Harmonic Series

The harmonic series is one of the first examples of an infinite series that diverges. It is expressed as \(\sum_{n=1}^{\infty} \frac{1}{n}\). Despite the terms \(\frac{1}{n}\) tending towards zero as \(n\) increases, the harmonic series grows without bound.

• This behavior is critical for comparison tests, as it serves as a known benchmark for divergence.
• The harmonic series is widely known in mathematics due to its simple form but surprising divergence.

In the given exercise, the series \(\sum \sin \left( \frac{1}{n} \right)\) was compared to this series, leading to the conclusion of divergence.

Infinite Series

An infinite series is the sum of an infinite sequence of terms. It can take the form \(\sum_{n=1}^{\infty} a_n\). Determining whether such a series converges or diverges is a major focus in mathematical analysis.

• Convergent series settle at a fixed number while divergent series grow indefinitely or oscillate without settling.
• The understanding of infinite series is key in mathematical fields such as calculus and analysis.

The exercise provided us with an infinite series involving a sinusoidal function, requiring a detailed investigation of its convergent nature.

Divergence

Divergence in the context of a series refers to the series not having a finite limit. When a series diverges, the sum either increases indefinitely or does not settle at a specific value.

• Divergent series do not satisfy finite limits, making them critical in understanding and manipulating infinite sums.
• Many tests, like the Limit Comparison Test, help determine the divergence or convergence of a series.

In our exercise, the series \(\sum \sin \left( \frac{1}{n} \right)\) was ultimately shown to diverge, similar to the harmonic series.

Sinusoidal Function

Sinusoidal functions, such as \(\sin(x)\), are periodic functions that oscillate between values. They are significant in various domains like physics, engineering, and mathematics.

• For small values of \(x\), \(\sin(x)\) behaves approximately like \(x\). This resemblance simplifies many calculations including those in series analysis.
• Understanding the behavior of sinusoidal functions is essential when involved in series, especially in determining convergence.

In this exercise, \(\sin \left( \frac{1}{n} \right)\) was approximated as \(\frac{1}{n}\), aiding in the application of the Limit Comparison Test.