Problem 14 - Thomas Calculus

Question

In Exercises \(9-16,\) use the Root Test to determine if each series converges absolutely or diverges. $$\sum_{n=1}^{\infty} \sin ^{n}\left(\frac{1}{\sqrt{n}}\right)$$

Step-by-Step Solution

Answer

The series converges absolutely.

Step 1: Understanding the Root Test

The Root Test is useful for assessing the convergence of a series \(\sum a_n\). For the series \(\sum a_n\), calculate \(\limsup_{n \to \infty} \sqrt[n]{|a_n|}\):- If the limit is less than 1, the series converges absolutely.- If the limit is greater than 1, the series diverges.- If the limit equals 1, the test is inconclusive.

Step 2: Identify the Series Terms

In this problem, our series is \(\sum_{n=1}^{\infty} \sin^n\left(\frac{1}{\sqrt{n}}\right)\). Here, each term \(a_n\) is given by \(\sin^n\left(\frac{1}{\sqrt{n}}\right)\).

Step 3: Apply the Root Test Formula

We need to compute \( \limsup_{n \to \infty} \sqrt[n]{|a_n|} \), where \(|a_n| = \left|\sin^n\left(\frac{1}{\sqrt{n}}\right)\right| = \sin^n\left(\frac{1}{\sqrt{n}}\right)\) because \(\sin^n\left(\frac{1}{\sqrt{n}}\right)\) is always positive.

Step 4: Calculate the \(n\)-th Root

Calculate \(\sqrt[n]{\sin^n\left(\frac{1}{\sqrt{n}}\right)} = \sin\left(\frac{1}{\sqrt{n}}\right)\).

Step 5: Evaluate the Limit Superior (\(\limsup\))

Evaluate \(\limsup_{n \to \infty} \sin\left(\frac{1}{\sqrt{n}}\right)\). Since \(\frac{1}{\sqrt{n}} \to 0\) as \(n \to \infty\), and using the continuity of \(\sin(x)\) around \(x=0\), we find that \(\sin\left(\frac{1}{\sqrt{n}}\right) \to \sin(0) = 0\). Thus, the \(\limsup\) is 0.

Step 6: Conclusion Based on the Root Test

Since \(\limsup_{n \to \infty} \sqrt[n]{|a_n|} = 0 < 1\), the series \(\sum_{n=1}^{\infty} \sin^n\left(\frac{1}{\sqrt{n}}\right)\) converges absolutely.

Key Concepts

Absolute Convergence

Absolute convergence is a stronger form of convergence that applies to series in mathematical analysis. A series \( \sum a_n \) is said to converge absolutely if the series of absolute values \( \sum |a_n| \) converges. This concept can help establish the convergence of a series even if some of its individual terms are negative.

When tackling problems involving series, determining absolute convergence can simplify the analysis, as it assures that rearranging the terms of the series will not affect the sum of the series. This property is a powerful tool in analysis because it provides stability in the behavior of series.

In our example, applying the Root Test involves calculating \( \limsup_{n \to \infty} \sqrt[n]{|a_n|} \). If the result is less than 1, the series converges absolutely, ensuring the series' overall convergence. This makes absolute convergence an essential check in the study of infinite series.

Series Convergence

Series convergence refers to the behavior of the sum of terms in a sequence as the number of terms grows indefinitely. Specifically, a series \( \sum a_n \) converges if the sequence of its partial sums \( S_n = a_1 + a_2 + \ldots + a_n \) approaches a finite limit as \( n \to \infty \).

To determine the convergence of a series, several tests including the Root Test can be applied. The Root Test involves evaluating the limit superior \( \limsup \) of the \( n \)-th root of the absolute value of the series terms. In our example, calculating \( \limsup_{n \to \infty} \sqrt[n]{\sin^n\left(\frac{1}{\sqrt{n}}\right)} \) revealed that it equals 0, establishing convergence.

Convergence is critical in mathematical analysis because it allows the approximation and understanding of infinite processes through finite sums. Identifying convergence is especially crucial in applications where stability and predictability of a series' behavior over infinite terms are required.

Limit Superior

The limit superior, denoted as \( \limsup \), is a concept used to capture the "supremum limit" or the greatest limit point of a sequence. This involves assessing the largest accumulation point a sequence approaches as it progresses.

Specifically, for a sequence \( \{a_n\} \), \( \limsup_{n \to \infty} a_n \) represents the highest value that \( a_n \) approaches infinitely often. Diverging from the typical notion of a limit that prioritizes average behavior, the \( \limsup \) accounts for oscillations and extremes in sequences.

In the context of the Root Test, applying \( \limsup \) to \( \sqrt[n]{|a_n|} \) overcomes the challenge of intermittent large values in a sequence that might suggest divergence. Thus, it provides a robust metric for judging the behavior of series terms, ultimately leading to meaningful insights into the convergence or divergence of the whole series. In our exercise, calculating \( \limsup_{n \to \infty} \sin\left(\frac{1}{\sqrt{n}}\right) \) led to the value 0, reinforcing the absolute convergence of the series.