Problem 14 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 13–30, classify each series as absolutely convergent, conditionally convergent, or divergent. $$ \sum_{n=1}^{\infty}(-1)^{n+1} \frac{1}{5 n^{1.1}} $$

Step-by-Step Solution

Answer

The series is absolutely convergent.

Step 1: Identify the series and its format

The series given is \( \sum_{n=1}^{\infty}(-1)^{n+1} \frac{1}{5 n^{1.1}} \). This is an alternating series, as indicated by the \((-1)^{n+1}\) factor, which changes the sign of each term.

Step 2: Check for absolute convergence

To determine absolute convergence, consider the series without the alternating factor: \( \sum_{n=1}^{\infty} \frac{1}{5n^{1.1}} \). This is a p-series where \( p = 1.1 \). A p-series is convergent if \( p > 1 \). Since \( 1.1 > 1 \), this p-series converges. Therefore, the series \( \sum_{n=1}^{\infty} \left| (-1)^{n+1} \frac{1}{5n^{1.1}} \right| \) is convergent.

Step 3: Consider the result of absolute convergence

Since the absolute value series \( \sum_{n=1}^{\infty} \frac{1}{5n^{1.1}} \) converges, the original alternating series \( \sum_{n=1}^{\infty}(-1)^{n+1} \frac{1}{5 n^{1.1}} \) is absolutely convergent. An absolutely convergent series is also considered conditionally convergent by default, but the absolute convergence is the stronger form.

Key Concepts

Alternating Series

An alternating series is a series where the signs of the terms alternate between positive and negative. This is usually indicated by a factor like \((-1)^{n+1}\) or \((-1)^n\) in the series expression. Such series might seem tricky, but they are an important part of calculus as they offer insights into convergence in situations where the series might not converge absolutely.
For a given alternating series \( \sum_{n=1}^{\infty}(-1)^n a_n \) with positive terms \(a_n \), we check if it converges using the Alternating Series Test. This test states that an alternating series converges if two conditions are met:

• The absolute value of the terms \(a_n\) decreases monotonically.
• The term \(a_n\) approaches zero as \(n\) approaches infinity.

These criteria might appear simple, but failing either one can impact convergence. Importantly, distinguishing alternating series helps identify when a more stringent convergence check, like absolute convergence, is unnecessary.

Absolute Convergence

Absolute convergence of a series occurs if the series of absolute values converges. Let's consider the series \( \sum_{n=1}^{\infty} (-1)^{n+1} \frac{1}{5 n^{1.1}} \). To test for absolute convergence, we look at the series without the alternating sign: \( \sum_{n=1}^{\infty} \left| \frac{1}{5 n^{1.1}} \right| \).
If the series of absolute values converges, we say the original series converges absolutely. Absolute convergence is a stronger condition than conditional convergence and gives us powerful tools for analyzing series:

• If a series converges absolutely, the terms can be rearranged in any order without affecting the sum.
• It also helps in determining the radius of convergence in power series.

Determining if a series is absolutely convergent can be crucial in analysis, as it often simplifies further computations and considerations.

p-Series

A p-series is a type of infinite series expressed in the form \( \sum_{n=1}^{\infty} \frac{1}{n^p} \). The convergence of a p-series depends on the value of \(p\):

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

In the exercise, we encounter the series \( \sum_{n=1}^{\infty} \frac{1}{5 n^{1.1}} \), which is a p-series where \(p = 1.1\). Since \(1.1 > 1\), this indicates that the p-series converges. Understanding p-series is crucial in calculus as they form building blocks for more complex series analyses. They serve as benchmarks or reference points for other series evaluations.

Convergent Series

A convergent series is one where the infinite sum approaches a finite limit. Convergence is fundamental in mathematical analysis as it indicates stability and finiteness in the series sum. For the series \( \sum_{n=1}^{\infty}(-1)^{n+1} \frac{1}{5 n^{1.1}} \), convergence is assessed by considering both the alternating nature and absolute convergence.
A series can be convergent in two primary ways:

• Absolutely Convergent: If the series of absolute values converges, the original series is also convergent by a stronger form.
• Conditionally Convergent: The series converges based solely on its alternating properties, not the absolute values.

In this case, as shown in the step-by-step solution, since the series \( \sum_{n=1}^{\infty} \frac{1}{5 n^{1.1}} \) converges, the original alternating series is absolutely convergent because when we ignore the alternating signs, the series still converges. Understanding the convergence of series is essential for deeper studies in mathematics, especially in topics related to power series, functions, and calculus applications.