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Power Series

Definition of Power Series

The series $\sum_{n=0}^{\infty} a_n x^n$ is called a power series, where $a_n$ are the coefficients and $x$ is the variable.

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
$a_n$	Mathematical symbol	Coefficient	Coefficient of the $n$ -th term in power series
$R$	Mathematical symbol	Radius of convergence	Radius where the power series converges

Convergence

Power Series Convergence

There exists a radius of convergence $R$ such that:

When $|x| < R$ , the series converges
When $|x| > R$ , the series diverges
When $|x| = R$ , convergence must be determined separately

The radius of convergence can be found using the ratio test or root test.

几何解释

推论

证明

符号说明

Methods to Find Radius of Convergence

Ratio Test

Ratio Test for Radius of Convergence

If $\lim_{n \to \infty} \left|\frac{a_{n+1}}{a_n}\right| = L$ , then:

When $L = 0$ , $R = +\infty$
When $L = +\infty$ , $R = 0$
When $0 < L < +\infty$ , $R = \frac{1}{L}$

几何解释

推论

证明

符号说明

Root Test

Root Test for Radius of Convergence

If $\lim_{n \to \infty} \sqrt[n]{|a_n|} = L$ , then:

When $L = 0$ , $R = +\infty$
When $L = +\infty$ , $R = 0$
When $0 < L < +\infty$ , $R = \frac{1}{L}$

几何解释

推论

证明

符号说明

Examples

Example 1

Find the radius of convergence of the power series $\sum_{n=0}^{\infty} \frac{x^n}{n!}$ .

Solution: $a_n = \frac{1}{n!}$

$\lim_{n \to \infty} \left|\frac{a_{n+1}}{a_n}\right| = \lim_{n \to \infty} \frac{n!}{(n+1)!} = \lim_{n \to \infty} \frac{1}{n+1} = 0$

Therefore, the radius of convergence $R = +\infty$ , meaning it converges for all real numbers $x$ .

Example 2

Find the radius of convergence of the power series $\sum_{n=0}^{\infty} x^n$ .

Solution: $a_n = 1$

$\lim_{n \to \infty} \left|\frac{a_{n+1}}{a_n}\right| = \lim_{n \to \infty} 1 = 1$

Therefore, the radius of convergence $R = 1$ , meaning it converges when $|x| < 1$ .

Example 3

Find the radius of convergence of the power series $\sum_{n=0}^{\infty} n! x^n$ .

Solution: $a_n = n!$

$\lim_{n \to \infty} \left|\frac{a_{n+1}}{a_n}\right| = \lim_{n \to \infty} \frac{(n+1)!}{n!} = \lim_{n \to \infty} (n+1) = +\infty$

Therefore, the radius of convergence $R = 0$ , meaning it converges only when $x = 0$ .

Example 4

Find the radius of convergence of the power series $\sum_{n=0}^{\infty} \frac{x^n}{n^2}$ .

Solution: $a_n = \frac{1}{n^2}$

$\lim_{n \to \infty} \left|\frac{a_{n+1}}{a_n}\right| = \lim_{n \to \infty} \frac{n^2}{(n+1)^2} = 1$

Therefore, the radius of convergence $R = 1$ , meaning it converges when $|x| < 1$ .

Exercises

Exercise 1

Find the radius of convergence of the power series $\sum_{n=0}^{\infty} n! x^n$ .

Reference Answer(3 个标签)

power seriesradius of convergenceratio test

Problem-solving approach: Use the ratio test to find the radius of convergence.

Detailed steps:

Identify the series type: $\sum_{n=0}^{\infty} n! x^n$ is a power series
Determine the coefficients: $a_n = n!$
Calculate the ratio: $\lim_{n \to \infty} \left|\frac{a_{n+1}}{a_n}\right| = \lim_{n \to \infty} \frac{(n+1)!}{n!} = \lim_{n \to \infty} (n+1) = +\infty$
Radius of convergence: $R = \frac{1}{+\infty} = 0$

Answer: The radius of convergence is $0$ , meaning it converges only when $x = 0$ .

Exercise 2

Find the radius of convergence of the power series $\sum_{n=0}^{\infty} \frac{x^n}{n^2}$ .

Reference Answer(3 个标签)

power seriesradius of convergenceratio test

Problem-solving approach: Use the ratio test to find the radius of convergence.

Detailed steps:

Identify the series type: $\sum_{n=0}^{\infty} \frac{x^n}{n^2}$ is a power series
Determine the coefficients: $a_n = \frac{1}{n^2}$
Calculate the ratio: $\lim_{n \to \infty} \left|\frac{a_{n+1}}{a_n}\right| = \lim_{n \to \infty} \frac{n^2}{(n+1)^2} = 1$
Radius of convergence: $R = \frac{1}{1} = 1$

Answer: The radius of convergence is $1$ , meaning it converges when $|x| < 1$ .

Exercise 3

Find the radius of convergence of the power series $\sum_{n=0}^{\infty} \frac{x^n}{n^3}$ .

Reference Answer(3 个标签)

power seriesradius of convergenceratio test

Problem-solving approach: Use the ratio test to find the radius of convergence.

Detailed steps:

Identify the series type: $\sum_{n=0}^{\infty} \frac{x^n}{n^3}$ is a power series
Determine the coefficients: $a_n = \frac{1}{n^3}$
Calculate the ratio: $\lim_{n \to \infty} \left|\frac{a_{n+1}}{a_n}\right| = \lim_{n \to \infty} \frac{n^3}{(n+1)^3} = 1$
Radius of convergence: $R = \frac{1}{1} = 1$

Answer: The radius of convergence is $1$ , meaning it converges when $|x| < 1$ .

Exercise 4

Find the radius of convergence of the power series $\sum_{n=0}^{\infty} \frac{x^n}{2^n}$ .

Reference Answer(3 个标签)

power seriesradius of convergenceratio test

Problem-solving approach: Use the ratio test to find the radius of convergence.

Detailed steps:

Identify the series type: $\sum_{n=0}^{\infty} \frac{x^n}{2^n}$ is a power series
Determine the coefficients: $a_n = \frac{1}{2^n}$
Calculate the ratio: $\lim_{n \to \infty} \left|\frac{a_{n+1}}{a_n}\right| = \lim_{n \to \infty} \frac{2^n}{2^{n+1}} = \frac{1}{2}$
Radius of convergence: $R = \frac{1}{\frac{1}{2}} = 2$

Answer: The radius of convergence is $2$ , meaning it converges when $|x| < 2$ .

Exercise 5

Find the radius of convergence of the power series $\sum_{n=0}^{\infty} \frac{x^n}{n!}$ .

Reference Answer(3 个标签)

power seriesradius of convergenceratio test

Problem-solving approach: Use the ratio test to find the radius of convergence.

Detailed steps:

Identify the series type: $\sum_{n=0}^{\infty} \frac{x^n}{n!}$ is a power series
Determine the coefficients: $a_n = \frac{1}{n!}$
Calculate the ratio: $\lim_{n \to \infty} \left|\frac{a_{n+1}}{a_n}\right| = \lim_{n \to \infty} \frac{n!}{(n+1)!} = \lim_{n \to \infty} \frac{1}{n+1} = 0$
Radius of convergence: $R = \frac{1}{0} = +\infty$

Answer: The radius of convergence is $+\infty$ , meaning it converges for all real numbers $x$ .

Summary

Symbols Used in This Article

Symbol	Type	Pronunciation/Explanation	Meaning in This Article
$x$	Mathematical symbol	Variable	Variable in the power series
$L$	Mathematical symbol	Limit value	Limit of ratio or root value
$\lim$	Mathematical symbol	Limit	Represents limit of sequence or function
$n!$	Mathematical symbol	Factorial	n factorial, $n! = n \times (n-1) \times \cdots \times 1$
$n!$	Mathematical symbol	Factorial	$n$ factorial, $n! = n \times (n-1) \times \cdots \times 1$

Chinese-English Glossary

Chinese Term	English Term	IPA Pronunciation	Explanation
幂级数	power series	/ˈpaʊə ˈsɪəriːz/	Series of the form $\sum_{n=0}^{\infty} a_n x^n$
收敛半径	radius of convergence	/ˈreɪdiəs əv kənˈvɜːdʒəns/	Radius $R$ where the power series converges
系数	coefficient	/kəʊɪˈfɪʃənt/	Coefficients $a_n$ of terms in power series
比值判别法	ratio test	/ˈreɪʃiəʊ test/	Method to determine convergence using ratio of consecutive terms
根值判别法	root test	/ruːt test/	Method to determine convergence using $n$ -th root
收敛	convergence	/kənˈvɜːdʒəns/	Partial sums sequence has a finite limit
发散	divergence	/daɪˈvɜːdʒəns/	Partial sums sequence has no finite limit
阶乘	factorial	/fækˈtɔːriəl/	$n! = n \times (n-1) \times \cdots \times 1$

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