Problem 63 - Thomas Calculus

Question

Neither the Ratio Test nor the Root Test helps with \(p\) -series. Try them on $$\sum_{n=1}^{\infty} \frac{1}{n^{p}}$$ and show that both tests fail to provide information about convergence.

Step-by-Step Solution

Answer

Both the Ratio Test and the Root Test are inconclusive for the convergence of a p-series.

Step 1: Understand the Ratio Test

First, recall the Ratio Test for the convergence of an infinite series. For a series \( \sum a_n \), the test involves evaluating the limit \[ \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|. \] If this limit is less than 1, the series converges absolutely. If the limit is greater than 1, the series diverges. If it equals 1, the test is inconclusive.

Step 2: Apply the Ratio Test to p-series

For the p-series \( \sum_{n=1}^{\infty} \frac{1}{n^p} \), we have \( a_n = \frac{1}{n^p} \) and \( a_{n+1} = \frac{1}{(n+1)^p} \). The ratio we need is: \[ \frac{a_{n+1}}{a_n} = \frac{\frac{1}{(n+1)^p}}{\frac{1}{n^p}} = \left( \frac{n}{n+1} \right)^p. \] Now, evaluate the limit as \( n \to \infty \): \[ \lim_{n \to \infty} \left( \frac{n}{n+1} \right)^p = \lim_{n \to \infty} \left( 1 - \frac{1}{n+1} \right)^p = 1. \] Since the limit is 1, the Ratio Test is inconclusive.

Step 3: Understand the Root Test

Next, recall the Root Test for the convergence of an infinite series. For a series \( \sum a_n \), the test involves evaluating the limit \[ \lim_{n \to \infty} \sqrt[n]{|a_n|}. \] If this limit is less than 1, the series converges absolutely. If the limit is greater than 1, the series diverges. If it equals 1, the test is inconclusive.

Step 4: Apply the Root Test to p-series

For the p-series \( \sum_{n=1}^{\infty} \frac{1}{n^p} \), \( a_n = \frac{1}{n^p} \). We need to evaluate: \[ \lim_{n \to \infty} \sqrt[n]{\frac{1}{n^p}} = \lim_{n \to \infty} \frac{1}{n^{p/n}}. \] Notice that \( p/n \to 0 \) as \( n \to \infty \), so the limit becomes \( 1 \), as \( n^{p/n} \to 1 \). Hence, the Root Test is also inconclusive because the limit equals 1.

Key Concepts

Convergence Tests

Convergence tests are vital in analyzing whether an infinite series converges or diverges. These tests help us understand the behavior of series by providing rules or limits we can check. The two main outcomes of convergence tests are:

• If a series converges, then adding its infinite terms results in a finite number.
• If a series diverges, adding its infinite terms never results in a finite sum.

Convergence tests use different strategies to determine a series' behavior. Importantly, they produce specific numerical criteria explaining when to expect convergence. However, it's crucial to remember that a few tests may sometimes be inconclusive.

Ratio Test

The Ratio Test is a practical tool for determining if a series converges absolutely. It works by examining the limit of the ratio of successive terms within a series. The steps are:

• Calculate the ratio of the next term to the current term: \( \left| \frac{a_{n+1}}{a_n} \right| \).
• Evaluate the limit of this ratio as \( n \to \infty \).

The Ratio Test says:

• If the limit is less than 1, the series converges absolutely.
• If the limit is greater than 1, the series diverges.
• If the limit is equal to 1, the test is inconclusive.

With the \(p\)-series, the limit equals 1, showing the Ratio Test cannot decide about its convergence.

Root Test

The Root Test offers another method for checking the convergence of a series. To apply this test:

• Calculate the \( n \)th root of the absolute value of the term: \( \sqrt[n]{|a_n|} \).
• Evaluate the limit of this expression as \( n \to \infty \).

According to the Root Test:

• The series converges absolutely if the limit is less than 1.
• The series diverges if the limit is more than 1.
• The test is inconclusive if the limit equals 1.

When applied to the \(p\)-series, \( \lim_{n \to \infty} \frac{1}{n^{p/n}} = 1 \), making the Root Test inconclusive for determining convergence of \(p\)-series.

Series Convergence

Series convergence is a fundamental concept in mathematics. It describes how the sum of an infinite series' terms approaches a finite limit. Understanding if a series converges is crucial because it forms the foundation for rigorous analysis in calculus.
For the specific case of \(p\)-series, we have \( \sum_{n=1}^{\infty} \frac{1}{n^p} \). We must use convergence tests specifically suited for \(p\)-series behavior, unlike the Ratio and Root Tests, which fall short. The well-known \(p\)-test for convergence states:

• The series converges if \( p > 1 \).
• The series diverges if \( p \leq 1 \).