I found the following formula here: Taylor Series of tan &#x2061;<!-- ⁡ --> x . Taylor

Jaycee Mathis

Jaycee Mathis

Answered question

2022-05-30

I found the following formula here: Taylor Series of tan x.
Taylor series of tan x:
tan x = n = 1 ( 1 ) n 1 2 2 n ( 2 2 n 1 ) B 2 n ( 2 n ) ! x 2 n 1

Answer & Explanation

extractumzz

extractumzz

Beginner2022-05-31Added 9 answers

The exponential generating function for the Bernoulli Numbers is
(1) n = 0 B n x n n ! = x e x 1
The even part of (1) is
(2) n = 0 B 2 n x 2 n ( 2 n ) ! = x 2 coth ( x 2 )
Since cot ( x ) = i coth ( i x ), by substituting x i x, we get
n = 0 ( 1 ) n B 2 n x 2 n ( 2 n ) ! = i x 2 coth ( i x 2 ) (3) = x 2 cot ( x 2 )
Therefore,
(4) cot ( x ) = n = 0 ( 1 ) n B 2 n 2 2 n x 2 n 1 ( 2 n ) !
Since tan ( x ) = cot ( x ) 2 cot ( 2 x ), we get
tan ( x ) = n = 0 ( 1 ) n B 2 n 2 2 n ( x 2 n 1 2 2 2 n 1 x 2 n 1 ) ( 2 n ) ! (5) = n = 1 ( 1 ) n 1 B 2 n ( 2 2 n 1 ) 2 2 n x 2 n 1 ( 2 n ) !

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