Question

Use the comparison test to determine whether the following series converge. $$ \sum_{n=1}^{\infty} \frac{\sin (1 / n)}{\sqrt{n}} $$

Short answer

The series \( \sum_{n=1}^{\infty} \frac{\sin(1/n)}{\sqrt{n}} \) converges by the comparison test.

Step-by-step solution

Identify the Series

We are given the series \( \sum_{n=1}^{\infty} \frac{\sin(1/n)}{\sqrt{n}} \). The task is to determine whether this infinite series converges or diverges using the comparison test.

Understand the Comparison Test

The Comparison Test states that if \( 0 \leq a_n \leq b_n \) for all \( n \) and the series \( \sum b_n \) converges, then \( \sum a_n \) also converges. Conversely, if \( \sum b_n \) diverges, then \( \sum a_n \) diverges as well.

Establish a Comparison

We know that \( \sin(x) \) is approximately \( x \) when \( x \) is close to zero. Therefore, \( \sin(1/n) \approx 1/n \) for large \( n \). Thus, we compare the original series to \( \sum_{n=1}^{\infty} \frac{1/n}{\sqrt{n}} = \sum_{n=1}^{\infty} \frac{1}{n^{3/2}} \).

Analyze the Comparison Series

The series \( \sum_{n=1}^{\infty} \frac{1}{n^{3/2}} \) is a p-series with \( p = 3/2 > 1 \), which means it converges.

Conclude with the Comparison Test

Since \( 0 \leq \frac{\sin(1/n)}{\sqrt{n}} \leq \frac{1/n}{\sqrt{n}} = \frac{1}{n^{3/2}} \) for large \( n \) and \( \sum_{n=1}^{\infty} \frac{1}{n^{3/2}} \) converges, by the comparison test, the original series \( \sum_{n=1}^{\infty} \frac{\sin(1/n)}{\sqrt{n}} \) also converges.

Key concepts

Comparison Test

The comparison test is a useful tool in determining the convergence or divergence of an infinite series. It relies on comparing a given series to a second, more familiar series whose convergence we already understand. The test works as follows:

• If you have two series, \( \sum a_n \) and \( \sum b_n \), with \( 0 \leq a_n \leq b_n \) for all \( n \), and if the series \( \sum b_n \) converges, then \( \sum a_n \) also converges.
• Conversely, if \( \sum b_n \) diverges, then \( \sum a_n \) diverges as well.

Using the comparison test effectively involves selecting a series \( \sum b_n \) that resembles \( \sum a_n \) and whose behavior is already known. This method simplifies the process of determining convergence without direct computation.

Infinite Series

An infinite series is the sum of the terms of an infinite sequence. Simply put, if you keep adding numbers and never stop, you have an infinite series. The key challenge with infinite series is determining whether adding up all its terms leads to a finite number (converges) or grows indefinitely (diverges).

• Convergence means that as you add more and more terms, the total sum approaches a specific value.
• Divergence indicates that as you add terms indefinitely, the sum doesn't settle towards a particular number; it either keeps increasing towards infinity or oscillates without settling.

Understanding infinite series is crucial because they appear in various mathematical contexts, such as calculus and even in real-world applications like calculating interest or modeling physics phenomena.

P-Series

A p-series is a special type of infinite series expressed in the form \( \sum_{n=1}^{\infty} \frac{1}{n^p} \). In p-series:

• The parameter \( p \) is a positive constant defining the behavior of the series.
• If \( p > 1 \), the p-series converges, meaning that the sum of its terms approaches a finite value.
• If \( p \leq 1 \), the p-series diverges, as the sum does not approach a finite value.

In the context of the original exercise, recognizing that \( \frac{1}{n^{3/2}} \) forms a p-series with \( p = 3/2 \) (> 1), confirms convergence. This insight is pivotal when applying the comparison test, as it offers a familiar baseline for determining the behavior of more complex series.

Convergence and Divergence

The terms convergence and divergence are fundamental in the study of series and sequences. Understanding these concepts helps in analyzing whether an infinite sum results in a meaningful number.

• Convergence: If the sum of the terms in an infinite series approaches a finite limit, we say the series converges. This outcome is desirable when working with series in mathematical analyses and applications.
• Divergence: If the sum doesn't settle to a finite value, the series diverges. This can mean that the total grows indefinitely, typically leading to an infinite value, or the sum could lack a pattern altogether.

Convergence and divergence play a critical role in mathematical discussions and real-world situations, such as determining whether an infinite time series in finance stabilizes over time.