Problem 9 - Thomas Calculus

Question

In Exercises \(9-16,\) use the Limit Comparison Test to determine if each series converges or diverges. $$ \begin{array}{l}{\sum_{n=1}^{\infty} \frac{n-2}{n^{3}-n^{2}+3}} \\\ {\left(\text {Hint} : \text { Limit Comparison with } \sum_{n=1}^{\infty}\left(1 / n^{2}\right)\right)}\end{array} $$

Step-by-Step Solution

Answer

The series \( \sum_{n=1}^{\infty} \frac{n-2}{n^{3}-n^{2}+3} \) converges by the Limit Comparison Test.

Step 1: Identify the Given Series

We need to analyze the convergence or divergence of the given series \( \sum_{n=1}^{\infty} \frac{n-2}{n^{3}-n^{2}+3} \).

Step 2: Identify the Series for Comparison

The problem suggests using the series \( \sum_{n=1}^{\infty} \frac{1}{n^{2}} \) for the Limit Comparison Test. This known series is a \( p \)-series with \( p = 2 \), which converges.

Step 3: Formulate the Limit for Comparison

The Limit Comparison Test involves taking the limit \( \lim_{n \to \infty} \frac{a_n}{b_n} \), where \( a_n = \frac{n-2}{n^3-n^2+3} \) and \( b_n = \frac{1}{n^2} \).

Step 4: Set up the Limit Expression

The expression becomes:\[\lim_{n \to \infty} \frac{\frac{n-2}{n^3-n^2+3}}{\frac{1}{n^2}} = \lim_{n \to \infty} \frac{n^2(n-2)}{n^3-n^2+3}\]

Step 5: Simplify the Expression

Distribute to get: \[\lim_{n \to \infty} \frac{n^3 - 2n^2}{n^3 - n^2 + 3}\]

Step 6: Evaluate the Limit

Divide each term in the numerator and denominator by \( n^3 \) to simplify:\[\lim_{n \to \infty} \frac{1 - \frac{2}{n}}{1 - \frac{1}{n} + \frac{3}{n^3}} = \lim_{n \to \infty} \frac{1 - \frac{2}{n}}{1} = 1\]

Step 7: Apply the Limit Comparison Test Result

Since the limit is finite and positive (specifically 1), and \( \sum_{n=1}^{\infty} \frac{1}{n^2} \) converges, the Limit Comparison Test tells us that \( \sum_{n=1}^{\infty} \frac{n-2}{n^{3}-n^{2}+3} \) also converges.

Key Concepts

Series Convergence

When examining whether a series converges or diverges, we are essentially exploring if the sum of its infinite terms approaches a specific number or goes off to infinity. A series converges if its partial sums eventually settle at a finite number as more terms are added. Understanding the convergence of series is crucial in mathematics, especially in calculus, as it helps predict the behavior of complex functions and is essential in various applications like physics and engineering.
To determine convergence, we often use tests like the Limit Comparison Test, which can help draw conclusions about a series' behavior by comparing it with another series whose convergence properties are known. This comparison can provide insights without directly summing an infinite number of terms, simplifying the problem considerably.

P-Series

A p-series is a particular type of series that is expressed in the form:

• \( \sum_{n=1}^{\infty} \frac{1}{n^p} \)

Here, \( p \) is a positive constant, and the convergence of the p-series depends entirely on the value of \( p \). If \( p > 1 \), the series converges. If \( p \leq 1 \), the series diverges.
The p-series \( \sum_{n=1}^{\infty} \frac{1}{n^2} \) is one of the most famous examples, known to converge because its exponent \( p = 2 \) is greater than 1. This knowledge is handy in the Limit Comparison Test, where a known convergent or divergent series, like a p-series, is used as a baseline to evaluate other series.

Limit Evaluation

Limit evaluation is a critical process in calculus that helps determine the behavior of sequences and series as they progress towards infinity. In the context of the Limit Comparison Test, we are often required to evaluate limits to ascertain the convergence of a series compared to another well-known series.
For the given problem, we evaluate the limit \[\lim_{n \to \infty} \frac{\frac{n-2}{n^3-n^2+3}}{\frac{1}{n^2}}\]by simplifying it to reach a manageable form. By simplifying the expression to \[\lim_{n \to \infty} \frac{n^2(n-2)}{n^3-n^2+3}\]and dividing by \( n^3 \), we can then further simplify to \[\lim_{n \to \infty} \frac{1 - \frac{2}{n}}{1 - \frac{1}{n} + \frac{3}{n^3}}\].
As \( n \) approaches infinity, the fractions with \( n \) in the denominator tend to zero, allowing us to conclude the limit equals 1, a positive finite number.

Calculus Concepts

Calculus is the mathematical study of change and motion, and it involves several key concepts, such as derivatives and integrals. In the context of series analysis, calculus aids in evaluating both series and the limits necessary for comparison tests.
The Limit Comparison Test, a widely used tool, transforms complex infinite series questions into more manageable problems by comparing them to simple, well-understood series like p-series.
Calculus techniques simplify these problems by reducing the series to forms whose behavior as \( n \to \infty \) can be easily analyzed. Through these calculus techniques, students connect the abstract nature of infinite sequences and series with practical analysis methods, enhancing their grasp of functions and real-world applications.
By understanding calculus concepts, learners can better predict and model continuous change, leading to deeper insights across various scientific and engineering disciplines.