How Is the Tan Power Series Derived Using Sin, Cos, and Bernoulli Numbers?

[Piano man]

How can the tan power series be derived from the sin and cos power series?
Where do the Bernoulli numbers come in?

 

[jackmell]

It's generally not done from the sine and cosine series:

http://www.mathhelpforum.com/math-help/f25/power-series-tangent-function-108861.html

 

[Piano man]

I was expecting something yucky, and this doesn't disappoint...

Thanks for the link :)

 

[ross_tang]

Piano man. Here is a link in http://www.voofie.com/concept/Mathematics/" that you maybe interested.

http://www.voofie.com/content/117/an-explicit-formula-for-the-euler-zigzag-numbers-updown-numbers-from-power-series/"

I derived the power series of the function sec x + tan x. For the tan x power series, you just take the odd terms from the expansion, since tan x is an odd function. While sec x corresponds to the even terms from the power series, as sec x is even.

It doesn't really show how Bernoulli numbers enter the expression, but it derives an explicit form for Bernoulli numbers.

 

[CellCoree]

The tan power series can be derived from the sin and cos power series by using the identity tan(x) = sin(x)/cos(x). This allows us to rewrite the tan function in terms of the sin and cos functions. By substituting this identity into the power series expansions of sin(x) and cos(x), we can manipulate the terms to obtain the power series for tan(x). This is a useful technique for expanding the tan function into a series of terms that can be more easily manipulated and approximated.

The Bernoulli numbers come into play when we expand the power series for cos(x) using the Maclaurin series formula, which involves the Bernoulli numbers. This is because the coefficients in the power series for cos(x) are related to the Bernoulli numbers through a specific formula. In this way, the Bernoulli numbers are indirectly involved in the derivation of the tan power series.

Moreover, the Bernoulli numbers also appear in the coefficients of the power series for tan(x) itself. This is because the power series for tan(x) involves only odd powers of x, and the Bernoulli numbers are involved in the coefficients of these odd powers. Therefore, the Bernoulli numbers play a crucial role in the derivation of the tan power series and are intimately linked to the coefficients of the series.