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Chapter 8: Problem 48

Use known facts about \(p\) -series to determine whether the given series converges or diverges. $$ \sum_{n=1}^{\infty} \frac{1}{n^{(\pi-e)}} $$

Short Answer

Expert verified
The series diverges because \( p = \pi-e < 1 \).

Step by step solution

01

Identify the p-series

The given series is \( \sum_{n=1}^{\infty} \frac{1}{n^{(\pi-e)}} \). Notice that it is a form of a p-series, which is generally of the form \( \sum_{n=1}^{\infty} \frac{1}{n^p} \). In this case, \( p = \pi-e \).
02

Determine the value of p

Calculate an approximation for \( \pi-e \) using known values: \( \pi \approx 3.14159 \) and \( e \approx 2.71828 \). Thus, \( \pi-e \approx 3.14159 - 2.71828 = 0.42331 \).
03

Apply the p-series test for convergence

The p-series test states that \( \sum_{n=1}^{\infty} \frac{1}{n^p} \) converges if \( p > 1 \) and diverges if \( p \leq 1 \). Since \( \pi-e \approx 0.42331 \) which is less than 1, the series does not meet the convergence criteria.
04

Conclusion: Analyze the result

Based on the p-series test, since \( \pi-e \leq 1 \), the series \( \sum_{n=1}^{\infty} \frac{1}{n^{(\pi-e)}} \) diverges.

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Key Concepts

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

Understanding the p-series
A p-series is a special type of infinite series that takes the form:
  • \( \sum_{n=1}^{\infty} \frac{1}{n^p} \)
In a p-series, the variable \( p \) is crucial to determining the behavior of the series.
The series converges or diverges based on the value of \( p \). If the series converges, it means the total sum approaches a fixed number, while divergence means the sum goes to infinity.
For example, the famous "harmonic series", \( \sum_{n=1}^{\infty} \frac{1}{n} \), is a p-series where \( p = 1 \). Despite seeming like a potential candidate for convergence due to its decreasing terms, it surprisingly diverges.
Exploring Convergence Tests
Determining whether a series converges or diverges is a fundamental question in calculus. There are various tests used to reach these conclusions, one of which is the p-series test.
  • The p-series test specifically applies to series of the form \( \sum_{n=1}^{\infty} \frac{1}{n^p} \).
  • According to this test, a series converges if \( p > 1 \).
  • If \( p \leq 1 \), the series diverges.

Another helpful tool is the "comparison test" which examines series by comparing them to a known convergent or divergent series. This provides additional reinforcement when analyzing series beyond p-series.
The Role of Calculus in Series Analysis
Calculus is a branch of mathematics that deals with continuous change, and it plays a significant role in analyzing infinite series.
At its core, calculus provides the tools to understand convergence and divergence through notions of limits and continuity.
  • The concept of limits helps evaluate the behavior of series as they extend towards infinity.
  • Integrals and derivatives, cornerstone concepts of calculus, offer insights into the accumulation and rate of change, respectively.
Understanding these concepts is essential for thoroughly analyzing series and effectively applying tests, such as the p-series test, to determine convergence.
Insights into Infinite Series
An infinite series is essentially the sum of infinitely many terms. These series are written in the form:
  • \( \sum_{n=1}^{\infty} a_n \)
The focus is on understanding whether adding an endless number of terms gives a finite number or not.
Key concepts related to infinite series include:
  • "Partial sums" which involve adding a finite number of terms to understand early patterns.
  • "Convergence" indicating the terms eventually lead to a specific value.
  • "Divergence" where the total keeps increasing without approaching a fixed limit.

Infinite series appear in many areas of mathematics and applied science, making their convergence properties essential for practical applications.

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