Problem 63 - Thomas Calculus

Question

Show that \(\sum_{n=2}^{\infty}\left((\ln n)^{q} / n^{p}\right)\) converges for \(-\infty< q < \infty\) and \(p > 1\) (Hint: Limit Comparison with \(\sum_{n=2}^{\infty} 1 / n^{r}\) for \(1 < r < p . )\)

Step-by-Step Solution

Answer

The series converges for any real \(q\) and \(p > 1\).

Step 1: Identify the Type of Series

The given series is \( \sum_{n=2}^{\infty} \frac{(\ln n)^{q}}{n^{p}} \). This is a series where each term consists of a function of \(n\) in the numerator and another in the denominator. We want to determine if this series converges.

Step 2: Understand the Comparison Series

The hint suggests using the series \( \sum_{n=2}^{\infty} \frac{1}{n^r} \) for \(1 < r < p\) as a reference point. This is a geometric series, which converges if \( r > 1 \).

Step 3: Use Limit Comparison Test

For the Limit Comparison Test, we need to compare the given series \( \sum_{n=2}^{\infty} \frac{(\ln n)^{q}}{n^{p}} \) with \( \sum_{n=2}^{\infty} \frac{1}{n^r} \). Define the terms as \( a_n = \frac{(\ln n)^q}{n^p} \) and \( b_n = \frac{1}{n^r} \).

Step 4: Evaluate the Limit

Evaluate the limit \( \lim_{n \to \infty} \frac{a_n}{b_n} = \lim_{n \to \infty} \frac{(\ln n)^q}{n^{p-r}} \). As \( n \) goes to infinity, if \( p > r \), then the exponent of \(n\) in the denominator is positive, ensuring the term goes to zero. For any constant \( C > 0 \), the product of this limit will converge given that the series \( \sum \frac{1}{n^r} \) converges.

Step 5: Apply the Convergence Result

Since \( \lim_{n \to \infty} \frac{(\ln n)^q}{n^{p-r}} = 0 \) for any real number \(q\) and since the reference series \( \sum_{n=2}^{\infty} \frac{1}{n^r} \) converges for \(1 < r < p\), the Limit Comparison Test implies that \( \sum_{n=2}^{\infty} \frac{(\ln n)^q}{n^p} \) converges.

Key Concepts

Limit Comparison Test

The Limit Comparison Test is a handy tool when you want to determine if a given series converges. The idea is to compare it with another series that you already know converges or diverges. By examining the limit of the ratio of their terms, you can reach a conclusion about the convergence of the original series.

Here's a step-by-step approach:

• Consider the series you want to test, which we'll call \( \sum a_n \).
• Choose another series \( \sum b_n \) that is similar, and check if \( b_n > 0 \) for all \( n \).
• Compute the limit \( \lim_{n \to \infty} \frac{a_n}{b_n} \).
• If the limit is a positive finite number, both series either converge or diverge together.

In our exercise, we compared \( \sum \frac{(\ln n)^q}{n^p} \) with the more familiar series \( \sum \frac{1}{n^r} \). Since the simpler comparison series converges for \( 1 < r < p \), we conclude convergence for our original series.

Geometric Series

A geometric series is one where each term is a constant multiple of the previous term. Consider a series of the form \( \sum a \, r^n \). It converges to a limit if the absolute value of \( r \) is less than 1.

Characteristics of a geometric series:

• The ratio \( r \) between consecutive terms is constant.
• An infinite geometric series \( \sum a \, r^n \) converges if \( |r| < 1 \).
• Its sum is \( a / (1 - r) \) for \( |r| < 1 \).

In the given exercise, the series \( \sum \frac{1}{n^r} \) is similar to a geometric series because its terms decrease at a constant exponential rate as \( n \) increases. This is why we chose it for the Limit Comparison Test to demonstrate convergence.

Mathematical Proofs

Mathematical proofs are structured arguments that establish the truth of a mathematical statement. They follow logical steps, based on accepted principles and definitions.

Steps in creating a proof:

• Clearly state what you need to prove.
• Use definitions, theorems, and logical reasoning to support your arguments.
• Structure your proof in a logical order, ensuring each step follows from the last.

In the context of our exercise, the proof involves demonstrating series convergence by examining the limit of the ratio of terms from two series. This methodical approach reassures us of the results through established mathematical principles.

Infinite Series

An infinite series is the sum of infinitely many terms. The series converges if the sum approaches a specific value as the number of terms goes to infinity. Otherwise, it diverges.

Important aspects of infinite series:

• A series like \( \sum a_n \) has partial sums \( S_n = a_1 + a_2 + \ldots + a_n \).
• The series converges if \( \lim_{n \to \infty} S_n \) approaches a finite number.
• If the partial sum diverges, the entire series does too.

The exercise focuses on understanding these concepts through the series \( \sum \frac{(\ln n)^q}{n^p} \), showing its convergence with the aid of the Limit Comparison Test. The careful balance of terms ensures that the sum approaches a finite limit, thus confirming convergence.