How do you find the power series for inverse trig functions? Can I find the power series for arcsin by manipulating the power series for sin? Thanks!
I don't think so. Even for the simplest of functions you already run into trouble, consider for example y = x^{2n} (for n = 1, 2, ...). The power series for x^{1/(2n)} is already non-trivial.
You can, it is called "reversion" of a series. But the formulas get more and more complicated as you proceed. For arcsin, a better way to find the series is to start with the binomial series for [itex] (1-x^2)^{-1/2}[/itex] and integrate term-by-term.
Identify the inverse trig function as a hypergeometric function, and manipulate the series expansion of the hypergeometric function. Any book on hypergeometric functions will give the necessary formulae.