Problem 22 - Thomas Calculus

Question

In Exercises \(17-46,\) use any method to determine if the series converges or diverges. Give reasons for your answer. $$ \sum_{n=1}^{\infty}\left(\frac{n-2}{n}\right)^{n} $$

Step-by-Step Solution

Answer

The series diverges because the limit of its general term is not zero.

Step 1: Analyze the General Term

First, consider the general term of the series: \( a_n = \left( \frac{n-2}{n} \right)^n \). Simplify \( \frac{n-2}{n} \) to \( 1 - \frac{2}{n} \). This gives \( a_n = \left( 1 - \frac{2}{n} \right)^n \).

Step 2: Use the Limit Definition

Examine the behavior of \( a_n \) as \( n \to \infty \). Recall that \( \lim_{n \to \infty} \left( 1 - \frac{x}{n} \right)^n = e^{-x} \). Applying this gives: \( \lim_{n \to \infty} \left( 1 - \frac{2}{n} \right)^n = e^{-2} \).

Step 3: Apply Divergence Test

The Divergence Test states that if \( \lim_{n \to \infty} a_n eq 0 \), the series \( \sum a_n \) diverges. Since \( \lim_{n \to \infty} a_n = e^{-2} eq 0 \), the series diverges.

Key Concepts

Divergence Test

The divergence test, also known as the nth term test, is a critical tool in determining whether a series converges or diverges. It involves analyzing the limit of a series' general term as it moves toward infinity. If the limit of the general term does not approach zero, the series is sure to diverge. This is because any series whose terms do not decay to zero cannot sum to a finite value, thus diverging by nature.

Here’s a simple breakdown of how the divergence test works:

• Identify the general term of the series, denoted as \( a_n \).
• Calculate \( \lim_{{n \to \infty}} a_n \).
• If \( \lim_{{n \to \infty}} a_n eq 0 \), then the series \( \sum a_n \) diverges.
• If \( \lim_{{n \to \infty}} a_n = 0 \), the test is inconclusive; further analysis is needed.

Remember, the divergence test can only a confirm divergence, but if the limit evaluates to zero, it does not guarantee convergence.

Limit of a Sequence

The concept of the limit of a sequence is vital in understanding the convergence behavior of series. A sequence \( a_n \) converges to a limit \( L \) if, as \( n \) approaches infinity, \( a_n \) gets arbitrarily close to \( L \). Formally, this means that for every positive number \( \epsilon \), there exists a natural number \( N \) such that for all \( n > N \), the distance between \( a_n \) and \( L \) is less than \( \epsilon \).

• Calculate \( \lim_{{n \to \infty}} a_n = L \).
• If the limit \( L \) exists and is finite, the sequence converges.
  If there's no limit or it's infinite, the sequence diverges.

This limits are fundamental when applying tests for series convergence as they provide evidence about the behavior of the series terms at infinity.

Infinite Series

An infinite series is a sum of terms that extends infinitely, usually represented as \( \sum_{n=1}^{\infty} a_n \). The study of infinite series is crucial in calculus because it allows us to approximate functions, analyze waveforms, and solve differential equations.

When considering an infinite series, we are interested in finding whether it converges to a finite sum or diverges to infinity. This depends on the behavior of the individual terms of the series:

• Convergence: The series adds up to a finite value as more terms are included.
• Divergence: The sum becomes infinite or fails to approach any particular value.
• Tests: There are various methods, such as the divergence test, ratio test, or integral test, used to determine the behavior of an infinite series.

Understanding the convergence or divergence of an infinite series has significant implications for both theoretical mathematics and practical applications.

General Term Analysis

In dealing with infinite series, the analysis of the general term \( a_n \) is a preliminary yet vital step. By examining \( a_n \), we can infer a wealth of information about the series' behavior.

• The first step often involves simplifying \( a_n \) if possible, breaking it down into elements that are easier to analyze, such as polynomials or rational expressions.
• Evaluate the limit of \( a_n \) as \( n \to \infty \) to apply tests of convergence.
• Look for recognizable patterns in \( a_n \) to relate it to known convergent or divergent series.

Such analysis determines what tests or theorems can be applied effectively to ascertain the series’ convergence or divergence. Detailed general term analysis forms the backbone of tackling any series analysis problem, guiding the choice of the most appropriate mathematical tests to employ.