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Chapter 11: Problem 22

Use comparison tests to determine whether the infinite series converge or diverge. \(\sum_{n=1}^{\infty} \frac{\arctan n}{n}\)

Short Answer

Expert verified
The series \(\sum_{n=1}^{\infty} \frac{\arctan n}{n}\) diverges.

Step by step solution

01

Understand the Series

The series in question is \(\sum_{n=1}^{\infty} \frac{\arctan n}{n}\). We need to determine if this series converges or diverges using comparison tests.
02

Select a Known Comparable Series

A good candidate for comparison is \(\sum_{n=1}^{\infty} \frac{1}{n}\), known as the harmonic series, which is known to diverge.
03

Determine the Behaviour of \(\arctan n\)

The function \(\arctan n\) is a non-negative function that increases and is bounded such that \(0 \leq \arctan n < \frac{\pi}{2}\) for all natural numbers \(n\).
04

Set Up the Comparison

We know that \(\arctan n \leq \frac{\pi}{2}\) for all \(n\). Therefore, \(\frac{\arctan n}{n} \leq \frac{\pi}{2n}\). This allows us to compare it with the series \(\sum_{n=1}^{\infty} \frac{1}{n}\).
05

Apply the Limit Comparison Test

Using the Limit Comparison Test, consider \(a_n = \frac{\arctan n}{n}\) and \(b_n = \frac{1}{n}\). Compute the limit: \[ \lim_{n \to \infty} \frac{a_n}{b_n} = \lim_{n \to \infty} \frac{\arctan n}{n} \times n = \lim_{n \to \infty} \arctan n = \frac{\pi}{2}.\]Since this limit is a positive finite number, both series have the same behavior.
06

Conclude from the Comparison

Since the harmonic series \(\sum_{n=1}^{\infty} \frac{1}{n}\) diverges and the limit comparison shows the two series behave similarly, the series \(\sum_{n=1}^{\infty} \frac{\arctan n}{n}\) also diverges.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Infinite Series
An infinite series is a sum of an infinite number of terms. For instance, the series \( \sum_{n=1}^{\infty} \frac{\arctan n}{n} \) is an example where the sum continues indefinitely. Infinite series are fundamental in mathematics because they allow us to explore and understand functions and sequences deeply.
  • The general form of an infinite series is \( \sum_{n=1}^{\infty} a_n \), where \( a_n \) represents the sequence of terms.
  • Determining whether an infinite series converges or diverges is crucial in various fields, from calculus to more advanced mathematical concepts.
A series is said to converge if it approaches a specific value as more terms are added, whereas it diverges if the sum continues to grow larger without bound. Understanding the behavior of an infinite series is essential in ensuring that mathematical manipulations of these series are valid and useful.
Harmonic Series
The harmonic series, represented as \( \sum_{n=1}^{\infty} \frac{1}{n} \), is one of the most well-known divergent series. Despite its simple form, the harmonic series is a fundamental example in the study of series and provides insight into the concept of divergence.
  • The harmonic series diverges, meaning that as you add more terms, the sum grows without bound.
  • It is often used as a comparison benchmark when analyzing other series.
Using the harmonic series, one can apply comparison tests to determine whether other series converge or diverge. By comparing a series to the harmonic series, you can often derive critical insights into its nature based on how similar the terms are to those of the harmonic series.
Limit Comparison Test
The Limit Comparison Test is a powerful tool used to determine the convergence or divergence of an infinite series by comparing it to another series with a known convergence behavior. It requires computing the limit of the ratio of the terms of the two series.
  • Select two sequences \( a_n \) and \( b_n \), where you know the behavior of one of the series \( \sum b_n \).
  • Compute the limit: \[ \lim_{n \to \infty} \frac{a_n}{b_n} \]
  • If the limit is a positive finite number, both series either converge or diverge together.
In our exercise, we used the Limit Comparison Test to compare \( \sum_{n=1}^{\infty} \frac{\arctan n}{n} \) with the harmonic series \( \sum_{n=1}^{\infty} \frac{1}{n} \). Since the limit yields a positive finite number, both series exhibit the same divergence behavior.
Convergence and Divergence
Convergence and divergence are central concepts in analyzing infinite series. Understanding whether a series converges or diverges helps determine the sum of the series and its usefulness in mathematical contexts.
  • A series converges if the sequence of its partial sums approaches a finite limit.
  • A series diverges if the partial sums do not approach a specific limit, potentially growing indefinitely.
In the context of the given series \( \sum_{n=1}^{\infty} \frac{\arctan n}{n} \), convergence and divergence were assessed using the Limit Comparison Test. Because this series behaves similarly to the divergent harmonic series, it, too, diverges. This conclusion is important for understanding the sum's behavior, which is essentially boundless.

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Most popular questions from this chapter

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