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p-Series

Definition

Definition of p-Series

The series $\sum_{n=1}^{\infty} \frac{1}{n^p}$ is called a p-series, where $p$ is a real number.

符号说明

Symbol	Type	Pronunciation/Explanation	Meaning in This Article
$\sum$	Greek letter	Sigma	Summation symbol, representing series
$\infty$	Mathematical symbol	Infinity	Represents infinite series, infinite number of terms
$p$	Mathematical symbol	Parameter	Parameter of p-series, determining convergence

Convergence

p-Series Convergence

The series converges when $p > 1$ and diverges when $p \leq 1$ .

几何解释

推论

证明

Let $f(x) = \frac{1}{x^p}$ , which is continuous, monotonically decreasing, and non-negative on $[1,+\infty)$ .
Compute the integral: when $p \neq 1$ , $\int_1^{+\infty} \frac{1}{x^p} dx = \frac{x^{1-p}}{1-p} \Big|_1^{+\infty}$
When $p > 1$ , the integral converges, implying the series converges; when $p < 1$ , the integral diverges, so the series diverges.
When $p = 1$ , $\int_1^{+\infty} \frac{1}{x} dx = \lim_{b\to\infty} \ln b = +\infty$ so the series diverges.

符号说明

Proof

Using the integral test:

Let $f(x) = \frac{1}{x^p}$ , then $f(x)$ is continuous, monotonically decreasing, and non-negative on $[1, +\infty)$ .

The convergence of the integral $\int_1^{+\infty} \frac{1}{x^p} dx$ :

When $p \neq 1$ : $\int_1^{+\infty} \frac{1}{x^p} dx = \frac{x^{1-p}}{1-p} \big|_1^{+\infty}$

When $p > 1$ , the integral converges
When $p < 1$ , the integral diverges

When $p = 1$ : $\int_1^{+\infty} \frac{1}{x} dx = \ln x \big|_1^{+\infty} = +\infty$

So the integral diverges.

Examples

Example 1: Determine the convergence of the series $\sum_{n=1}^{\infty} \frac{1}{n^2}$ .

Solution: This is a p-series with $p = 2 > 1$

Therefore, the series converges.

Example 2: Determine the convergence of the series $\sum_{n=1}^{\infty} \frac{1}{n^3}$ .

Solution: This is a p-series with $p = 3 > 1$

Therefore, the series converges.

Example 3: Determine the convergence of the series $\sum_{n=1}^{\infty} \frac{1}{\sqrt{n}}$ .

Solution: This is a p-series with $p = \frac{1}{2} \leq 1$

Therefore, the series diverges.

Exercises

Exercise 1

Determine the convergence of the series $\sum_{n=1}^{\infty} \frac{1}{n^3}$ .

Reference Answer(2 个标签)

p-seriesseries convergence

Problem-solving approach: This is a p-series; we need to determine the relationship between p and 1.

Detailed steps:

Identify the series type: $\sum_{n=1}^{\infty} \frac{1}{n^3}$ is a p-series
Determine the p value: $p = 3$
Check convergence: $p = 3 > 1$ , so the series converges

Answer: The series converges.

Exercise 2

Determine the convergence of the series $\sum_{n=1}^{\infty} \frac{1}{n^4}$ .

Reference Answer(2 个标签)

p-seriesseries convergence

Problem-solving approach: This is a p-series; we need to determine the relationship between p and 1.

Detailed steps:

Identify the series type: $\sum_{n=1}^{\infty} \frac{1}{n^4}$ is a p-series
Determine the p value: $p = 4$
Check convergence: $p = 4 > 1$ , so the series converges

Answer: The series converges.

Exercise 3

Determine the convergence of the series $\sum_{n=1}^{\infty} \frac{1}{\sqrt{n}}$ .

Reference Answer(2 个标签)

p-seriesseries convergence

Problem-solving approach: This is a p-series; we need to determine the relationship between p and 1.

Detailed steps:

Identify the series type: $\sum_{n=1}^{\infty} \frac{1}{\sqrt{n}} = \sum_{n=1}^{\infty} \frac{1}{n^{1/2}}$ is a p-series
Determine the p value: $p = \frac{1}{2}$
Check convergence: $p = \frac{1}{2} \leq 1$ , so the series diverges

Answer: The series diverges.

Exercise 4

Determine the convergence of the series $\sum_{n=1}^{\infty} \frac{1}{n^{1.5}}$ .

Reference Answer(2 个标签)

p-seriesseries convergence

Problem-solving approach: This is a p-series; we need to determine the relationship between p and 1.

Detailed steps:

Identify the series type: $\sum_{n=1}^{\infty} \frac{1}{n^{1.5}}$ is a p-series
Determine the p value: $p = 1.5$
Check convergence: $p = 1.5 > 1$ , so the series converges

Answer: The series converges.

Summary

Symbols Used in This Article

Symbol	Type	Pronunciation/Explanation	Meaning in This Article
$n$	Mathematical symbol	Number of terms	Number of terms in the series
$\int$	Mathematical symbol	Integral	Represents definite or indefinite integral
$\ln$	Mathematical symbol	Natural logarithm	Natural logarithm function
$\lim$	Mathematical symbol	Limit	Represents limit of sequence or function

Chinese-English Glossary

Chinese Term	English Term	IPA Pronunciation	Explanation
$p$ 级数	$p$ -series	/piː ˈsɪəriːz/	Series of the form $\sum_{n=1}^{\infty} \frac{1}{n^p}$
收敛	convergence	/kənˈvɜːdʒəns/	Partial sums sequence has a finite limit
发散	divergence	/daɪˈvɜːdʒəns/	Partial sums sequence has no finite limit
积分判别法	integral test	/ˈɪntɪɡrəl test/	Method to determine series convergence using integrals
比较判别法	comparison test	/kəmˈpærɪsən test/	Method to determine series convergence by comparison

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