I've been working through these listed Maclaurin series and deriving all of them. For the listed trigonometric functions, I'm wondering if there are ways to do things:
- not have Bernoulli or Euler numbers in the series
- find the series without having to do polynomial division
- $\sec x$
- $\operatorname{sech} x$
- $\tan x$
- $\tanh x$
Additionally, on the list of mathematical series for wikipedia, the following also have Bernoulli numbers:
- $\cot z$
- $\csc z$
- $\coth z$
- $\operatorname{csch} z$
I'm sure I could find them by doing division, but is there a better way? (And by "better way", I mean a connection between other series, like how $\sinh x$ can be made up of variations of $e^x$, or how $\sin^{-1} x$ can be found by manipulating binomial series)
In an earlier question of mine (Two versions of Maclaurin series for $\arcsin(x)$), I learned that there's a way to not have the gamma function and so now I'm able to find that series -- but is there a way to not have these Bernoulli or Euler numbers (and is that way understandable at a single-variable calculus level?)