Kuzee

Cosine Series

Definition

Definition of Cosine Series

The series ∑n=0∞(−1)n(2n)!x2n\sum_{n=0}^{\infty} \frac{(-1)^n}{(2n)!} x^{2n} is called the cosine series.

SymbolTypePronunciation/ExplanationMeaning in This Article
∑\sumGreek letterSigmaSummation symbol, representing series
∞\inftyMathematical symbolInfinityRepresents infinite series, infinite number of terms
n!n!Mathematical symbolFactorialn factorial, n!=n×(n−1)×⋯×1n! = n \times (n-1) \times \cdots \times 1

Convergence

Convergence of Cosine Series

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

∑n=0∞(−1)n(2n)!x2n=cos⁡x\sum_{n=0}^{\infty} \frac{(-1)^n}{(2n)!} x^{2n} = \cos x

证明

Use the ratio test:

an+1an=(−1)n+1(2(n+1))!x2(n+1)⋅(2n)!(−1)nx2n=−x2(2n+1)(2n+2)\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→∞∣an+1an∣=lim⁡n→∞x2(2n+1)(2n+2)=0<1\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 xx, the series converges.

Examples

Example 1

Find the sum of the series ∑n=0∞(−1)n(2n)!(π3)2n\sum_{n=0}^{\infty} \frac{(-1)^n}{(2n)!} \left(\frac{\pi}{3}\right)^{2n}.

Solution: This is a cosine series with x=π3x = \frac{\pi}{3}

So the sum is: cos⁡(π3)=12\cos\left(\frac{\pi}{3}\right) = \frac{1}{2}

Example 2

Find the sum of the series ∑n=0∞(−1)n(2n)!π2n\sum_{n=0}^{\infty} \frac{(-1)^n}{(2n)!} \pi^{2n}.

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

So the sum is: cos⁡(π)=−1\cos(\pi) = -1

Exercises

Exercise 1

Find the sum of the series ∑n=0∞(−1)n(2n)!(π4)2n\sum_{n=0}^{\infty} \frac{(-1)^n}{(2n)!} \left(\frac{\pi}{4}\right)^{2n}.

Reference Answer
cosine seriesseries summation

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

Detailed steps:

  1. Identify the series type: ∑n=0∞(−1)n(2n)!(π4)2n\sum_{n=0}^{\infty} \frac{(-1)^n}{(2n)!} \left(\frac{\pi}{4}\right)^{2n} is a cosine series
  2. Determine the parameter: x=π4x = \frac{\pi}{4}
  3. Compute the sum: S=cos⁡(π4)=22S = \cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}

Answer: The sum is 22\frac{\sqrt{2}}{2}.

Exercise 2

Find the sum of the series ∑n=0∞(−1)n(2n)!(π6)2n\sum_{n=0}^{\infty} \frac{(-1)^n}{(2n)!} \left(\frac{\pi}{6}\right)^{2n}.

Reference Answer
cosine seriesseries summation

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

Detailed steps:

  1. Identify the series type: ∑n=0∞(−1)n(2n)!(π6)2n\sum_{n=0}^{\infty} \frac{(-1)^n}{(2n)!} \left(\frac{\pi}{6}\right)^{2n} is a cosine series
  2. Determine the parameter: x=π6x = \frac{\pi}{6}
  3. Compute the sum: S=cos⁡(π6)=32S = \cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2}

Answer: The sum is 32\frac{\sqrt{3}}{2}.

Summary

Symbols Used in This Article

SymbolTypePronunciation/ExplanationMeaning in This Article
xxMathematical symbolVariableVariable in cosine series
π\piGreek letterPiCircular constant, approximately 3.14159
cos⁡\cosMathematical symbolCosineCosine function
lim⁡\limMathematical symbolLimitRepresents limit of sequence or function

Chinese-English Glossary

Chinese TermEnglish TermIPA PronunciationExplanation
余弦级数cosine series/ˈkəʊsaɪn ˈsɪəriːz/Series of the form ∑n=0∞(−1)n(2n)!x2n\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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