Problem 25 - Calculus Early Transcendentals: Pearson New International Edition

Question

Determine convergence or divergence for each of the series. Indicate the test you use. \(1+\frac{1}{2 \sqrt{2}}+\frac{1}{3 \sqrt{3}}+\frac{1}{4 \sqrt{4}}+\cdots\)

Step-by-Step Solution

Answer

The series converges by the P-Series Test.

Step 1: Analyze the General Term

The series given is \(\sum_{n=1}^{\infty} \frac{1}{n\sqrt{n}}\). The general term of the series is \( a_n = \frac{1}{n\sqrt{n}} \).

Step 2: Apply the P-Series Test

The P-Series Test states that a series \( \sum_{n=1}^{\infty} \frac{1}{n^p} \) converges if \( p > 1 \), and diverges if \( p \leq 1 \). In the given series, the term is \( \frac{1}{n^{3/2}} \). Here, \( p = \frac{3}{2} > 1 \).

Step 3: Determine Convergence

Since \( p = \frac{3}{2} > 1 \), the series \( \sum_{n=1}^{\infty} \frac{1}{n\sqrt{n}} \) converges according to the P-Series Test.

Key Concepts

P-Series Test

The P-Series Test is a powerful tool for analyzing series of the form \( \sum_{n=1}^{\infty} \frac{1}{n^p} \). Its primary purpose is to determine whether such a series converges or diverges. The core rule of the P-Series Test is:

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

In the case of the series \( \sum_{n=1}^{\infty} \frac{1}{n\sqrt{n}} \), it can be rewritten as \( \sum_{n=1}^{\infty} \frac{1}{n^{3/2}} \). Here, the exponent \( p = \frac{3}{2} \) is clearly greater than 1, so by the P-Series Test, we conclude that the series converges. This insight provides a straightforward way to handle series with similar forms.

General Term

The general term of a series is an important aspect to determine its properties and behavior. It represents the formula that generates the terms of the series as they appear. In our original exercise, the series given is \( \sum_{n=1}^{\infty} \frac{1}{n\sqrt{n}} \), and the general term is represented as \( a_n = \frac{1}{n\sqrt{n}} \).Recognizing the general term helps in applying convergence tests effectively. For instance, transforming \( a_n \) into a form that matches known test formats like the P-Series is crucial. This transformation often simplifies the step to conclude whether the series will converge or diverge.

Convergence Tests

Convergence tests are mathematical tools used to examine if a series converges (settles to a finite limit) or diverges (heads towards infinity or doesn't settle). They are pivotal in analyzing the behavior of series and include several different approaches:

• P-Series Test: Used specifically for sequences of the form \( \sum_{} \frac{1}{n^p} \).
• Comparison Test: Compares a problematic series to a simpler series with known convergence behavior.
• Ratio Test: Handles series featuring factorials or exponential terms.
• Integral Test: Relates a series to an improper integral for analysis.

Each test has its own set of conditions and is suited for different types of series formations. Choosing the correct test helps streamline the process of finding out a series' convergence status efficiently.

Series Analysis

Series analysis involves scrutinizing a series to understand its sum and behavior. To decide whether a series converges or diverges, you must start with identifying its general term and then choosing an appropriate convergence test. This task involves:

• Identifying Patterns: Look at the general term's growth or decay.
• Rewriting Terms: Simplify the series to enable easier application of tests, such as converting to P-Series form.
• Applying Tests: Use convergence tests like the P-Series Test to make decisions about the series.

In our specific example, recognizing that \( a_n = \frac{1}{n\sqrt{n}} \) could be rewritten to fit the P-Series allowed us to directly apply the P-Series Test. This approach made the process quick and efficient.