Power series of inverse trig functions

[zeebo17]

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!

 

[CompuChip]

I don't think so.
Even for the simplest of functions you already run into trouble, consider for example y = x²ⁿ (for n = 1, 2, ...). The power series for x^1/(2n) is already non-trivial.

 

[g_edgar]

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 (1-x^2)^{-1/2} and integrate term-by-term.

 

[MichaelMi]

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.