Problem 52 - Thomas Calculus

Question

Which of the series converge, and which diverge? Use any method, and give reasons for your answers. \begin{equation}\sum_{n=1}^{\infty} \frac{\sqrt[n]{n}}{n^{2}}\end{equation}

Step-by-Step Solution

Answer

The series \( \sum_{n=1}^{\infty} \frac{\sqrt[n]{n}}{n^{2}} \) converges.

Step 1: Identify the series

We are given the series \( \sum_{n=1}^{\infty} \frac{\sqrt[n]{n}}{n^{2}} \). Our task is to determine if this series converges or diverges.

Step 2: Simplify the General Term

The general term of our series is \( a_n = \frac{\sqrt[n]{n}}{n^{2}} \). The expression \( \sqrt[n]{n} \) can be rewritten using exponents as \( n^{1/n} \), leading to the term becoming \( \frac{n^{1/n}}{n^2} \).

Step 3: Estimate Limit of Simplified Term

As \( n \) approaches infinity, the term \( n^{1/n} \) approaches 1 because any number raised to power \(1/n \) approaches 1. Therefore, \( \lim_{n \to \infty} n^{1/n} = 1 \).

Step 4: Analyze the Behavior of the Term

Substituting the limit found in the previous step, the term \( a_n \) becomes \( \frac{1}{n^2} \) for large \( n \).

Step 5: Use the p-Series Test

The series \( \sum_{n=1}^{\infty} \frac{1}{n^2} \) is a p-series with \( p = 2 > 1 \). According to the p-series convergence test, a series \( \sum_{n=1}^{\infty} \frac{1}{n^p} \) converges if \( p > 1 \).

Step 6: Conclusion on Convergence

Since the series \( \sum_{n=1}^{\infty} \frac{1}{n^2} \) converges, and the general term of our series behaves like \( \frac{1}{n^2} \) for large \( n \), the series \( \sum_{n=1}^{\infty} \frac{\sqrt[n]{n}}{n^{2}} \) converges as well.

Key Concepts

Understanding the p-series test

The p-series test is a fundamental method used in determining the convergence or divergence of infinite series. It's especially useful when dealing with series of the form \( \sum_{n=1}^{\infty} \frac{1}{n^p} \). Here, the key variable is \( p \), known as the power of the series.

• If \( p > 1 \), the series converges. This means that as we sum up an infinite number of terms, the total approaches a finite number.
• If \( p \leq 1 \), the series diverges. This means as we continue to add terms from 1 to infinity, the total sum goes to infinity.

Finding \( p \) and understanding this simple rule allows us to quickly determine if a series like \( \sum_{n=1}^{\infty} \frac{1}{n^2} \) converges. In our specific scenario, since \( p = 2 > 1 \), we confirmed the series converges. Remember, the p-series test is a reliable strategy when the general term is a simple power of \( n \). Using this test provides clear insight when tackling series, and it's a powerful tool in an analyst's toolkit.

Exploring the limit comparison test

The limit comparison test is another crucial method for analyzing series. It helps determine the convergence of a particular series by comparing it to another series whose convergence behavior is already known. Here's how the test works:

• Start with two series, \( \sum a_n \) and \( \sum b_n \), whose terms are positive.
• Calculate the limit \( L = \lim_{n \to \infty} \frac{a_n}{b_n} \).
• If \( 0 < L < \infty \), then either both series converge, or both diverge. This means both series have the same behavior concerning convergence.

This test is particularly handy when a series' terms can be complex, and it is difficult to apply tests like the p-series directly. By comparing with a more "well-understood" series, conclusions can be drawn about the original series. For instance, if you can relate a complicated series back to a simpler one like \( \sum \frac{1}{n^2} \), and find \( L \) is finite and positive, you can confirm they share the same convergence properties.

Understanding infinite series

An infinite series is a sum of an infinite sequence of numbers. It's an essential concept in mathematics with numerous applications in calculus, physics, and engineering. Understanding the basics can simplify approaches to complex mathematical challenges.An infinite series is written as \( \sum_{n=1}^{\infty} a_n \), where \( a_n \) represents the individual terms.

Types of Series

• Convergent Series: These have a sum that approaches a finite number as more terms are added. The prime example is the p-series with \( p > 1 \).
• Divergent Series: These series sum to infinity or do not settle to a specific value. For instance, the harmonic series \( \sum_{n=1}^{\infty} \frac{1}{n} \).

Recognizing the nature of an infinite series is pivotal for mathematical analysis. Convergent series are used to solve problems in various fields due to their predictable behavior. On the other hand, divergent series often indicate physical phenomena that are unstable or unbounded, requiring different math tools to manage or interpret. This understanding enables experts to distinguish between usable and unusable results in both theoretical and applied mathematics.