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Chapter 9: Problem 34

Determine whether the statement is true or false. Explain your answer. The series $$\sum_{k=1}^{\infty} \frac{1}{p^{k}}$$ is a \(p\) -series.

Short Answer

Expert verified
False, it is a geometric series, not a \( p \)-series.

Step by step solution

01

Understand the Given Series

The given series is \( \sum_{k=1}^{\infty} \frac{1}{p^{k}} \), where each term is \( \frac{1}{p^k} \). This is actually a geometric series because the series has the common ratio \( r = \frac{1}{p} \).
02

Recall the Definition of a p-series

A \( p \)-series is defined as \( \sum_{k=1}^{\infty} \frac{1}{k^{p}} \) where \( p \) is a constant. In a \( p \)-series, the variable \( k \) is in the base and \( p \) is in the exponent, unlike the given series.
03

Compare the Given Series to a p-series

In the given series \( \sum_{k=1}^{\infty} \frac{1}{p^{k}} \), the constant \( p \) is part of the base and raised to a power equal to the position \( k \). This is different from a \( p \)-series where the variable \( k \) is the base and \( p \) is a constant in the denominator.
04

Determine the Truth Value of the Statement

Since the given series does not match the form of a \( p \)-series, as explained in the previous steps, the statement that \( \sum_{k=1}^{\infty} \frac{1}{p^{k}} \) is a \( p \)-series is false.

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Key Concepts

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

Understanding p-series
A p-series is a type of infinite series formatted as \( \sum_{k=1}^{\infty} \frac{1}{k^{p}} \). Here, the denominator involves the variable \( k \) raised to the power \( p \), where \( p \) is a constant.
Common characteristics include:
  • If \( p > 1 \), the p-series converges, meaning it approaches a finite sum.
  • If \( p \leq 1 \), the p-series diverges, meaning it grows without bounds or in other words, doesn't settle to a finite value.
Understanding whether a series is a p-series or not is crucial in determining its convergence or divergence.
The given series \( \sum_{k=1}^{\infty} \frac{1}{p^{k}} \) does not align with the format of a p-series, as the variable \( k \) is in the exponent rather than the base.
Nature of Infinite Series
Infinite series involve summing an infinite sequence of terms, usually in the form of \( \sum_{n=1}^{\infty} a_n \).
The behavior of these series can vary significantly:
  • Some infinite series sum up to a finite value or converge.
  • Others may diverge, indicating that they lack a unique sum as they extend to infinity.
These series play a key role in various mathematical and practical applications, such as approximating functions and calculating areas.
Not every infinite series can be easily classified as convergent or divergent; it often requires specific tests like the comparison test, ratio test, or integral test.
Convergence and Divergence
Convergence and divergence are central concepts when working with infinite series.
A series is said to converge if the sum of its terms approaches a specific number. Conversely, a series diverges if it fails to approach any limit.
Determining convergence or divergence often depends on analyzing the terms of the series:
  • The convergence of a series can sometimes be assessed using the Comparison Test, Ratio Test, or Integral Test, each of which provides a way to evaluate the behavior of a series as the number of terms increases.
  • In the example of a geometric series \( \sum_{k=1}^{\infty} \frac{1}{p^{k}} \), convergence depends on the common ratio \( r \), specifically, if \( |r| < 1 \).
  • A p-series, on the other hand, converges when \( p > 1 \).
Understanding these principles helps in exploring and explaining the results of series analysis.

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Most popular questions from this chapter

Consider the series $$\sum_{k=1}^{\infty} \frac{(-3)^{k}}{k} x^{k}$$ Determine the intervals of convergence for this series and for the series obtained by integrating this series term by term.

(a) Find an interval \([0, b]\) over which \(e^{x}\) can be approximated by \(1+x+\left(x^{2} / 2 !\right)\) to three decimal-place accuracy throughout the interval. (b) Check your answer in part (a) by graphing $$\left|e^{x}-\left(1+x+\frac{x^{2}}{2 !}\right)\right|$$ over the interval you obtained.

Read about Zeno's dichotomy paradox in an appropriate reference work and relate the paradox in a setting that is familiar to you. Discuss a connection between the paradox and geometric series.

Determine whether the series converges. $$\sum_{k=1}^{\infty}\left(1+\frac{1}{k}\right)^{-k}$$

Show that the nth Taylor polynomial for sinh \(x\) about \(x=\ln 4\) is $$\sum_{k=0}^{n} \frac{16-(-1)^{k}}{8 k !}(x-\ln 4)^{k}$$
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