Cosine Series

Definition

Definition of Cosine Series

The series $\sum_{n=0}^{\infty} \frac{(-1)^n}{(2n)!} x^{2n}$ is called the cosine 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 of Cosine Series

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

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

证明

Use the ratio test:

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

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

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

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

Solution: This is a cosine series with $x = \frac{\pi}{3}$

So the sum is: $\cos\left(\frac{\pi}{3}\right) = \frac{1}{2}$

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

Solution: This is a cosine series with $x = \pi$

So the sum is: $\cos(\pi) = -1$

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

Reference Answer (2 个标签)

cosine series series summation

Problem-solving approach: This is a cosine series; identify the value of $x$ .

Detailed steps:

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

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

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

Reference Answer (2 个标签)

cosine series series summation

Problem-solving approach: This is a cosine series; identify the value of $x$ .

Detailed steps:

Identify the series type: $\sum_{n=0}^{\infty} \frac{(-1)^n}{(2n)!} \left(\frac{\pi}{6}\right)^{2n}$ is a cosine series
Determine the parameter: $x = \frac{\pi}{6}$
Compute the sum: $S = \cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2}$

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

Symbol	Type	Pronunciation/Explanation	Meaning in This Article
$x$	Mathematical symbol	Variable	Variable in cosine series
$\pi$	Greek letter	Pi	Circular constant, approximately 3.14159
$\cos$	Mathematical symbol	Cosine	Cosine function
$\lim$	Mathematical symbol	Limit	Represents limit of sequence or function

Chinese Term	English Term	IPA Pronunciation	Explanation
余弦级数	cosine series	/ˈkəʊsaɪn ˈsɪəriːz/	Series of the form $\sum_{n=0}^{\infty} \frac{(-1)^n}{(2n)!} x^{2n}$
收敛	convergence	/kənˈvɜːdʒəns/	Partial sums sequence has a finite limit
比值判别法	ratio test	/ˈreɪʃiəʊ test/	Method to determine convergence using ratio of consecutive terms

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