Question

Testing for Convergence of \(p\) -series For each of the following series, determine whether it converges or diverges. a. \(\sum_{n=1}^{\infty} \frac{1}{n^{4}}\) b. \(\sum_{n=1}^{\infty} \frac{1}{n^{2 / 3}}\)

Short answer

a. Converges; b. Diverges.

Step-by-step solution

Understanding a p-series

A p-series is defined as \( \sum_{n=1}^{\infty} \frac{1}{n^p} \). The convergence or divergence of this series depends on the value of \( p \). Specifically, the series converges if \( p > 1 \) and diverges if \( 0 < p \leq 1 \).

Analyze the first series for convergence

The series given is \( \sum_{n=1}^{\infty} \frac{1}{n^4} \). Here, \( p = 4 \). Since \( p > 1 \), this series converges according to the convergence criteria for p-series.

Analyze the second series for convergence

The series given is \( \sum_{n=1}^{\infty} \frac{1}{n^{2/3}} \). Here, \( p = \frac{2}{3} \). Since \( p \leq 1 \), this series diverges according to the convergence criteria for p-series.

Key concepts

p-series

A p-series is a particular type of infinite series that takes the form \( \sum_{n=1}^{\infty} \frac{1}{n^p} \), where \( p \) is a positive constant. These series are significant in mathematical analysis as they provide a straightforward means to test for convergence and divergence. The behavior of a p-series heavily depends on the value of \( p \). If \( p > 1 \), the series will converge. This means its terms approach zero at a rate fast enough for the sum to settle towards a particular value. On the other hand, if \( 0 < p \leq 1 \), the series diverges, implying that the sum increases indefinitely without approaching a finite limit. Understanding this distinction is essential as it forms the basis for deeper explorations into series and their behaviors.

divergence

Divergence in the context of infinite series means that the series does not approach any finite limit. If a series diverges, the sum of its terms becomes infinitely large or oscillates without settling into any value. For p-series, divergence occurs when the power \( p \) is less than or equal to 1. Take, for example, the series \( \sum_{n=1}^{\infty} \frac{1}{n^{2/3}} \). Here, \( p = \frac{2}{3} \), which is less than 1, leading to divergence.

• The series terms decrease too slowly, as \( n \) grows, to sum up neatly.
• Divergent series cannot be meaningfully summed to a single value.

The purpose of identifying divergence is often to understand the limitations of series and whether or not they can be practically utilized in mathematical models or calculations.

infinite series

An infinite series is a sequence of numbers summed indefinitely. It is denoted by the expression \( \sum_{n=1}^{\infty} a_n \), where \( a_n \) represents the terms of the series. Infinite series play a crucial role in calculus and analysis for modeling continuous growth, decay, and other processes. Successful assessment of an infinite series depends on whether it converges or diverges.

• If the series converges, it approaches a particular value, adding more terms still brings you towards that value.
• If it diverges, it either continually adds up to infinity or becomes erratic.

The study and application of infinite series rely on analyzing their behavior, using various tests to determine whether they sum to a finite number or not.

convergence criteria

Convergence criteria are the rules and tests that help determine whether an infinite series converges or diverges. When analyzing a series such as a p-series, these criteria help decide whether the series is useful for practical applications or mathematical insights. For a p-series, the criteria for convergence are clear:

• Converges if \( p > 1 \), indicating a sufficiently rapid decrease in terms.
• Diverges if \( 0 < p \leq 1 \), where the decrease in terms is insufficient for a finite sum.

These criteria ensure that a series reaching a sum is predictable, allowing mathematicians to ground their analyses and computations on consistent properties. Knowing these rules is fundamental for working with series, especially in scenarios where series need to be summed or used for approximating more complex functions.