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.