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Sine Series

Definition

Sine Series Definition

The series $\sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)!} x^{2n+1}$ is called the sine series.

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
$n!$	Mathematical symbol	Factorial	n factorial, $n! = n \times (n-1) \times \cdots \times 1$

Convergence

Sine Series Convergence

For any real number $x$ , the series converges, and its sum is:

$\sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)!} x^{2n+1} = \sin x$

证明

Using the ratio test:

$\frac{a_{n+1}}{a_n} = \frac{(-1)^{n+1}}{(2(n+1)+1)!} x^{2(n+1)+1} \cdot \frac{(2n+1)!}{(-1)^n x^{2n+1}} = -\frac{x^2}{(2n+2)(2n+3)}$

$\lim_{n \to \infty} \left|\frac{a_{n+1}}{a_n}\right| = \lim_{n \to \infty} \frac{x^2}{(2n+2)(2n+3)} = 0 < 1$

Therefore, for any real number $x$ , the series converges.

Examples

Example 1

Find the sum of the series $\sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)!} \left(\frac{\pi}{2}\right)^{2n+1}$ .

Solution: This is the sine series with $x = \frac{\pi}{2}$ .

Therefore, the sum is: $\sin\left(\frac{\pi}{2}\right) = 1$

Example 2

Find the sum of the series $\sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)!} \pi^{2n+1}$ .

Solution: This is the sine series with $x = \pi$ .

Therefore, the sum is: $\sin(\pi) = 0$

Exercises

Exercise 1

Find the sum of the series $\sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)!} \left(\frac{\pi}{6}\right)^{2n+1}$ .

Reference Answer(2 个标签)

sine seriesseries summation

Solution Approach: This is the sine series, need to determine the value of $x$ .

Detailed Steps:

Identify series type: $\sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)!} \left(\frac{\pi}{6}\right)^{2n+1}$ is the sine series
Determine parameter: $x = \frac{\pi}{6}$
Calculate sum: $S = \sin\left(\frac{\pi}{6}\right) = \frac{1}{2}$

Answer: The sum is $\frac{1}{2}$ .

Exercise 2

Find the sum of the series $\sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)!} \left(\frac{\pi}{3}\right)^{2n+1}$ .

Reference Answer(2 个标签)

sine seriesseries summation

Solution Approach: This is the sine series, need to determine the value of $x$ .

Detailed Steps:

Identify series type: $\sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)!} \left(\frac{\pi}{3}\right)^{2n+1}$ is the sine series
Determine parameter: $x = \frac{\pi}{3}$
Calculate sum: $S = \sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}$

Answer: The sum is $\frac{\sqrt{3}}{2}$ .

Exercise 3

Find the sum of the series $\sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)!} \left(\frac{\pi}{4}\right)^{2n+1}$ .

Reference Answer(2 个标签)

sine seriesseries summation

Solution Approach: This is the sine series, need to determine the value of $x$ .

Detailed Steps:

Identify series type: $\sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)!} \left(\frac{\pi}{4}\right)^{2n+1}$ is the sine series
Determine parameter: $x = \frac{\pi}{4}$
Calculate sum: $S = \sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}$

Answer: The sum is $\frac{\sqrt{2}}{2}$ .

Summary

Symbols Appearing in This Article

Symbol	Type	Pronunciation/Explanation	Meaning in This Article
$x$	Mathematical symbol	Variable	Variable in the sine series
$\pi$	Greek letter	Pi	Pi, approximately 3.14159
$\sin$	Mathematical symbol	Sine	Sine function
$\lim$	Mathematical symbol	Limit	Represents the limit of a sequence or function

Chinese-English Glossary

Chinese Term	English Term	Phonetic	Explanation
Sine series	sine series	/saɪn ˈsɪəriːz/	Series of the form $\sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)!} x^{2n+1}$
Convergence	convergence	/kənˈvɜːdʒəns/	Series partial sums have a finite limit
Ratio test	ratio test	/ˈreɪʃiəʊ test/	Method to determine convergence using ratio of adjacent terms

Trigonometric Series

Learn sine series, cosine series, tangent series, arctangent series, and other trigonometric function series.

Cosine Series

Learn the definition, convergence, and applications of cosine series.