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Tangent Series

Definition

Definition of Tangent Series

The series $\sum_{n=1}^{\infty} \frac{B_{2n} (-4)^n (1-4^n)}{(2n)!} x^{2n-1}$ is called the tangent series, where $B_{2n}$ are Bernoulli numbers.

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
$B_{2n}$	Mathematical symbol	Bernoulli numbers	Bernoulli numbers, used in tangent series expansion
$n!$	Mathematical symbol	Factorial	n factorial, $n! = n \times (n-1) \times \cdots \times 1$

Convergence

Convergence of Tangent Series

Interval of convergence: $(-\frac{\pi}{2}, \frac{\pi}{2})$
The sum is:

$\sum_{n=1}^{\infty} \frac{B_{2n} (-4)^n (1-4^n)}{(2n)!} x^{2n-1} = \tan x$

几何解释

推论

证明

符号说明

Explanation

The coefficients of the tangent series involve Bernoulli numbers, making its form relatively complex. In practical applications, the first few terms are usually used for approximation.

Summary

Symbols Used in This Article

Symbol	Type	Pronunciation/Explanation	Meaning in This Article
$B_{2n}$	Mathematical symbol	Bernoulli numbers	Bernoulli numbers, used in tangent series expansion
$x$	Mathematical symbol	Variable	Variable in tangent series
$\pi$	Greek letter	Pi	Circular constant, approximately 3.14159
$\tan$	Mathematical symbol	Tangent	Tangent function

Chinese-English Glossary

Chinese Term	English Term	IPA Pronunciation	Explanation
正切级数	tangent series	/ˈtændʒənt ˈsɪəriːz/	Series expansion of the tangent function
伯努利数	Bernoulli numbers	/bɜːˈnuːli ˈnʌmbəz/	An important sequence in number theory
收敛区间	interval of convergence	/ˈɪntəvəl əv kənˈvɜːdʒəns/	Interval where the series converges
收敛	convergence	/kənˈvɜːdʒəns/	Partial sums sequence has a finite limit

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