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Chapter 9: Problem 42

p-Series Define a p-series and state the requirements for its convergence.

Short Answer

Expert verified
A p-series is a series of the form \( \sum_{n=1}^{\infty} \frac{1}{n^p} \), where n is a positive integer, and p is a real number. This series converges if p > 1 and diverges if p 鈮� 1.

Step by step solution

01

Definition of a p-Series

A p-series is a series of the form \( \sum_{n=1}^{\infty} \frac{1}{n^p} \). Here, n is a positive integer, and p is a real number.
02

Requirement for Convergence - p > 1

Convergence of a p-series is decided by the value of p. The p-series \( \sum_{n=1}^{\infty} \frac{1}{n^p} \) converges if p > 1.
03

Requirement for Divergence - p 鈮� 1

The p-series \( \sum_{n=1}^{\infty} \frac{1}{n^p} \) diverges if p 鈮� 1.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Calculus and Infinite Series
Calculus is a branch of mathematics focusing on limits, functions, derivatives, integrals, and infinite series. An infinite series is a sequence of numbers that are added together indefinitely. The concept of the infinite series is pivotal in calculus, as it helps in understanding the behavior of functions over an unlimited domain.

When we talk about an infinite series, we often want to know if the sum of its terms converges to a particular value or if it diverges, meaning it lacks a finite limit. The study of these series and their properties is an essential part of calculus, as it connects to many other concepts such as integral and differential equations.
Understanding p-Series
A p-series is an important type of infinite series in calculus. It is represented by the expression \( \sum_{n=1}^{\infty} \frac{1}{n^p} \), where \( n \) is a positive integer and \( p \) is a real number. The series gets its name from the variable \( p \) that appears as the exponent in the denominator.

Understanding the behavior of p-series is vital because they serve as a benchmark for comparing the convergence or divergence of other series through various comparison tests. They are also related to the Riemann zeta function, which has implications in number theory and complex analysis.
Convergence Criteria for Series
To determine whether an infinite series converges, we use specific convergence criteria. In the context of a p-series, the convergence depends directly on the value of \( p \).

  • If \( p > 1 \), the p-series converges. This means the sequence of partial sums of the series approaches a specific number as \( n \) becomes infinitely large.
  • If \( p \leq 1 \), the series does not converge, meaning it diverges. So, it doesn't sum up to a finite limit as \( n \) grows without bounds.
These rules form the basis for one of the simplest convergence tests in calculus, known as the p-series test.
Divergence Criteria for Series
While convergence criteria help identify series that sum to finite values, divergence criteria are equally important as they highlight series that fail to converge to a limit. A p-series provides a clear-cut divergence criterion: if the exponent \( p \leq 1 \), the series diverges.

Other divergent behaviors include oscillation, where terms do not approach a single value, or growth without bound. For series in which the divergence is not immediately clear, there are tests, such as the nth-term test for divergence, which provide a methodological approach to determining a series' long-term behavior.

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Most popular questions from this chapter

Approximating an Integral In Exercises \(63-70\) , use a power series to approximate the value of the integral with an error of less than \(0.0001 .\) (In Exercises 65 and \(67,\) assume that the integrand is defined as 1 when \(x=0 .\) $$ \int_{0.1}^{0.3} \sqrt{1+x^{3}} d x $$

Finding a Series Explain how to use the series $$g(x)=e^{x}=\sum_{n=0}^{\infty} \frac{x^{n}}{n !}$$ to find the series for each function. Do not find the series. (a) \(f(x)=e^{-x}\) (b) \(f(x)=e^{3 x}\) (c) \(f(x)=x e^{x}\)

Finding a Taylor Series In Exercises \(1-12,\) use the definition of Taylor series to find the Taylor series, centered at \(c,\) for the function. $$ f(x)=\ln x, \quad c=1 $$

Using a Binomial Series In Exercises \(17-26,\) use the binomial series to find the Maclaurin series for the function. $$ f(x)=\frac{1}{\sqrt{1-x}} $$

Find the values of \(x\) for which the series converges. $$\sum_{n=0}^{\infty} 3(x-4)^{n}$$
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