Tangent Series

Definition

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.

$\sum$ (sigma): summation symbol.

$\infty$ (infinity): infinitely many terms.

$B_{2n}$ : Bernoulli numbers.

$n!$ : factorial, $n! = n \times (n-1) \times \cdots \times 1$ .

Convergence of the tangent series

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

Coefficients involve Bernoulli numbers and are intricate; in practice, we use the first few terms for approximation.

符号	类型	读音/说明	在本文中的含义
$\sum$	希腊字母	Sigma（西格玛）	求和符号，表示级数
$\infty$	数学符号	无穷大	表示无穷级数，项数无限
$B_{2n}$	数学符号	伯努利数	伯努利数，用于正切级数展开
$n!$	数学符号	阶乘	$n$ 的阶乘， $n! = n \times (n-1) \times \cdots \times 1$
$x$	数学符号	变量	正切级数中的变量
$\pi$	希腊字母	Pi（派）	圆周率，约等于 3.14159
$\tan$	数学符号	正切	正切函数

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