Problem 17 - Calculus Early Transcendentals: Pearson New International Edition

Question

Determine convergence or divergence for each of the series. Indicate the test you use. $$ \sum_{n=1}^{\infty} \frac{4 n^{3}+3 n}{n^{5}-4 n^{2}+1} $$

Step-by-Step Solution

Answer

The series converges by the Limit Comparison Test with \( \sum \frac{1}{n^2} \).

Step 1: Identify the Series Type

We have the series \( \sum_{n=1}^{\infty} \frac{4n^3 + 3n}{n^5 - 4n^2 + 1} \). This is an infinite series with rational expressions in terms of \( n \).

Step 2: Compare Leading Terms for Dominance

To analyze the behavior of the series, we'll compare the degrees of the polynomial in the numerator and the denominator. The leading term in the numerator is \( 4n^3 \) and in the denominator, it's \( n^5 \).

Step 3: Asymptotic Analysis

For large \( n \), higher-degree terms dominate lower-degree ones. Thus, \( \frac{4n^3}{n^5} \approx \frac{4}{n^2} \). This suggests that for large \( n \), the series behaves similarly to \( \sum \frac{4}{n^2} \).

Step 4: Apply the Limit Comparison Test

Compare \( a_n = \frac{4n^3 + 3n}{n^5 - 4n^2 + 1} \) with \( b_n = \frac{1}{n^2} \). Compute \( \lim_{n \to \infty} \frac{a_n}{b_n} = \lim_{n \to \infty} \frac{4n^3 + 3n}{n^5 - 4n^2 + 1} \times n^2 = \lim_{n \to \infty} \frac{4n^5 + 3n^3}{n^5 - 4n^2 + 1} = 4 \).

Step 5: Determine Convergence by the Result of the Test

Since the limit in the Limit Comparison Test is a positive finite number (4), and \( \sum \frac{1}{n^2} \) is a convergent p-series (\(p = 2\)), \( \sum \frac{4n^3 + 3n}{n^5 - 4n^2 + 1} \) also converges.

Key Concepts

Infinite Series

An infinite series is the sum of an infinite sequence of numbers. Essentially, it involves adding terms endlessly, which can either approach a certain finite value or continue to grow without bound. The most common representation is using the summation notation, such as \( \sum_{n=1}^{\infty} a_n \), where \( a_n \) represents the terms in the series.
A series is said to be convergent if the sequence of its partial sums approaches a fixed number as more terms are added. If the sums do not stabilize to any number, the series is divergent.

• A convergent series leads to a specific value, meaning the infinite "sum" settles on a finite number.
• A divergent series either increases indefinitely or doesn't settle on any specific number.

Understanding whether a series converges or diverges is crucial in mathematical analysis, as it determines whether computations involve finite or infinite results. Various tests, like the Comparison Test or Ratio Test, help us determine the behavior of a series.

Limit Comparison Test

The Limit Comparison Test is a handy tool for analyzing the convergence of infinite series. It works by comparing a given series to another series whose convergence behavior is known. The idea is simple: if you have a complicated series, see if it behaves similarly to a simpler series.
Here's how the Limit Comparison Test is applied:

• Identify the target series \( \sum a_n \).
• Choose a simpler comparison series \( \sum b_n \) that resembles \( \sum a_n \) in behavior.
• Calculate the limit \( L = \lim_{n \to \infty} \frac{a_n}{b_n} \).

If \( L \) is a positive finite number, both series \( \sum a_n \) and \( \sum b_n \) either converge or diverge together. The test is particularly useful for dealing with series that have terms expressed as polynomials or fractions. It removes the complexity by reducing it to a more familiar series type.

p-Series Convergence

A p-series is a specific type of series where each term is of the form \( \frac{1}{n^p} \). The series \( \sum_{n=1}^{\infty} \frac{1}{n^p} \) has distinct convergence behaviors according to the value of \( p \):

• If \( p > 1 \), the p-series converges.
• If \( p \leq 1 \), the p-series diverges.

This fundamental behavior makes p-series an excellent baseline for comparison tests. In the given problem, the series \( \sum \frac{1}{n^2} \) is convergent since \( p = 2 \), which is greater than 1.
The simplicity of the p-series makes them easy to analyze and apply in various convergence tests. It's widely used in mathematical analysis, calculus, and even in fields like physics and engineering where understanding infinite processes is crucial. By linking the given series to the behavior of a p-series, we can simplify and resolve complex series to assess their convergence swiftly.