Problem 22 - Calculus: Early Transcendentals

Question

Determine whether the series converges or diverges. \( \displaystyle \sum_{n = 3}^{\infty} \frac {n + 2}{(n + 1)^3} \)

Step-by-Step Solution

Answer

The series converges by the Limit Comparison Test.

Step 1: Write out the General Term

The general term of the series is given by \( a_n = \frac{n + 2}{(n + 1)^3} \). Our goal is to determine whether this series converges or diverges as \( n \) approaches infinity.

Step 2: Apply the Limit Comparison Test

To determine convergence, we compare with a simpler series. Let's compare with the series \( \sum_{n=3}^{\infty} \frac{1}{n^2} \), a known convergent p-series with \( p = 2 \). Compute \( \lim_{n\to\infty} \frac{a_n}{b_n} \) where \( b_n = \frac {1}{n^2} \).

Step 3: Simplify the Comparison Ratio

Calculate the ratio:\[\lim_{n\to\infty} \frac{\frac{n + 2}{(n + 1)^3}}{\frac{1}{n^2}} = \lim_{n\to\infty} \frac{(n + 2) n^2}{(n + 1)^3}\]Simplifying this gives:\[\lim_{n\to\infty} \frac{n^3 + 2n^2}{n^3 + 3n^2 + 3n + 1}\]

Step 4: Evaluate the Limit

Dividing all terms by \( n^3 \), the limit becomes:\[\lim_{n \to \infty} \frac{1 + \frac{2}{n}}{1 + \frac{3}{n} + \frac{3}{n^2} + \frac{1}{n^3}}\]As \( n \to \infty \), this simplifies to \( 1 \).

Step 5: Conclude Using Limit Comparison Test Result

Since the limit of the ratio is a positive finite number (1), and the series \( \sum \frac{1}{n^2} \) converges, the series \( \sum \frac{n + 2}{(n + 1)^3} \) also converges by the Limit Comparison Test.

Key Concepts

Limit Comparison Test

The Limit Comparison Test is an essential tool for determining the convergence or divergence of an infinite series. It allows us to compare a complex series to a simpler, well-known series to draw conclusions about its behavior.

Here's how it works: if you have two series, \(\sum a_n\) and \(\sum b_n\), and you know whether \(\sum b_n\) converges or diverges, then you can use the Limit Comparison Test to determine the same for \(\sum a_n\). You do this by computing \(\lim_{n\to\infty} \frac{a_n}{b_n}\).

When the limit \(L\) is a positive finite number (meaning \(0 < L < \infty\)), both series either converge or diverge together. This test is particularly useful for series where the terms \(a_n\) can be closely approximated by a simpler series \(b_n\). In our example, the series \(\sum \frac{1}{n^2}\) is chosen because it behaves similarly to \(\frac{n + 2}{(n + 1)^3}\) for large \(n\). Since \(\sum \frac{1}{n^2}\) converges, and our comparison yielded \(L = 1\), we conclude that the original series converges.

p-series Convergence

A p-series is a specific type of series in the form \(\sum \frac{1}{n^p}\), where \(n\) is a positive integer and \(p\) is a real number. Understanding the behavior of p-series is fundamental because they serve as a basis for more complex comparison tests.

Here's the rule for p-series:

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

This straightforward test can often be applied directly or used in conjunction with other tests, such as the Limit Comparison Test, to determine the convergence or divergence of more complicated series.

In the exercise, the series \(\sum \frac{1}{n^2}\) serves as our basis for comparison. Since \(p = 2\), which is greater than 1, it indicates that the p-series converges, granting you confidence to apply this rule in different contexts.

Infinite Series Analysis

Analyzing an infinite series involves understanding the behavior of the series as the number of terms approaches infinity. This analysis draws on various convergence tests, such as the Limit Comparison Test and p-series convergence, all aimed at determining whether the series approaches a finite sum or not.

Infinite series are particularly interesting because of their diverse applications in mathematics, physics, and engineering. They can model real-world phenomena or describe functions in a concise way.

• The convergence of a series implies that adding increasingly many terms will settle to a fixed value.
• Divergence indicates that the series does not approach any finite limit as you sum more terms.

In our specific problem, the series \(\sum \frac{n + 2}{(n + 1)^3}\) is analyzed through careful comparison with the simpler p-series \(\sum \frac{1}{n^2}\). Through this method, we determined that our series converges. This type of systematic approach is integral to mastering infinite series analysis.