Problem 11 - Calculus: Early Transcendentals

Question

Determine whether the series is convergent or divergent. \( 1 + \frac {1}{8} + \frac {1}{27} + \frac {1}{64} + \frac {1}{125} + \cdot \cdot \cdot \)

Step-by-Step Solution

Answer

The series is convergent.

Step 1: Recognize the series terms pattern

First, note that the given series is composed of terms of the form \( \frac{1}{n^3} \) where \( n \) is an integer starting at 1. The terms can be written as follows: \[ a_n = \frac{1}{n^3} \] The series can, therefore, be written as:\[ \sum_{n=1}^{\infty} \frac{1}{n^3} \]

Step 2: Determine the type of series

Recognize that this is a p-series of the form \( \sum_{n=1}^{\infty} \frac{1}{n^p} \). In our case, the series is \( \sum_{n=1}^{\infty} \frac{1}{n^3} \), where \( p = 3 \).

Step 3: Apply p-series convergence test

For a p-series \( \sum_{n=1}^{\infty} \frac{1}{n^p} \), the series converges if \( p > 1 \) and diverges if \( p \leq 1 \). In this instance, we have \( p = 3 \), which is greater than 1, satisfying the condition for convergence.

Step 4: Conclude the behavior of the series

Since the condition \( p > 1 \) for the convergence of a p-series is satisfied (where \( p = 3 \)), the series \( \sum_{n=1}^{\infty} \frac{1}{n^3} \) is convergent.

Key Concepts

P-series

A p-series is a type of series where each term is of the form \( \frac{1}{n^p} \) with \( n \) being a positive integer starting from 1. These series are called p-series because the exponent "p" in the denominator crucially affects their behavior. When you encounter a p-series, identifying the value of "p" is essential for determining convergence or divergence.

• If \( p > 1 \), the series converges, meaning the sum of all terms up to infinity approaches a finite number.
• If \( p \leq 1 \), the series diverges, which implies that the sum grows infinitely as you add more terms.

For example, in the series \( \sum_{n=1}^{\infty} \frac{1}{n^3} \), \( p = 3 \). Since 3 is greater than 1, the series will converge. Understanding p-series is a critical foundational step in analyzing many other types of series.

Convergence Tests

When dealing with series, determining whether a series converges or diverges is key. Convergence tests are tools developed to help mathematicians and students assess the behavior of series.
One of the most common tests is the p-series test, specifically designed for series of the form \( \sum_{n=1}^{\infty} \frac{1}{n^p} \). If \( p > 1 \), it confirms convergence; if \( p \leq 1 \), divergence.
There are other tests available, such as:

• Comparison Test: Compares the given series with a known convergent or divergent series.
• Ratio Test: Uses the ratio of successive terms to find convergence or divergence.
• Integral Test: Relies on the integration of a related function for analysis.

Each test has specific scenarios where it's most applicable, helping streamline the process of analyzing the series. Recognizing when to use each test is crucial in the context of series analysis.

Infinite Series

An infinite series is essentially a sum of an infinite sequence of numbers. You're adding up numbers forever! These series can often behave in surprising ways, sometimes yielding a finite sum, and other times not stabilizing at all.
Infinite series appear frequently in mathematical analysis, physics, and engineering, representing real-world phenomena and functions that need such extensive summation. The number of terms is infinite, yet the outcome might still be a finite value or infinite.

• Convergent Infinite Series: These add up to a specific number as you approach infinity.
• Divergent Infinite Series: These do not settle into any finite sum, their total sum just keeps getting larger.

Understanding the behavior of infinite series often involves applying concepts like convergence tests. These tests allow determining whether the particular arrangement of numbers will eventually stop changing significantly, settling into a sum, or keep growing. They are fundamental in making sense of many mathematical and practical applications dealing with sums.