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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 $p \leq 1$ .

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.

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-series series 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-series series 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-series series 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-series series 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

Previous Geometric Series Learn the definition, convergence, and applications of geometric series.

Next Harmonic Series Learn the definition, divergence, and proof methods of the harmonic series.

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