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 series series 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 series series 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 series series 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

NextCosine SeriesLearn the definition, convergence, and applications of cosine series.

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