On Wed, 20 Feb 2013 19:04:46 -0500, Dave Rudolf <firstname.lastname@example.org> wrote:
> Hey folks, > > I was looking at the taylor series that is commonly used to compute > arc-tangent, which is > > atan(x) = x - x^3/3 + x^5/5 - x^7/7 + x^9/9.... > > However, the domain for arc-tangent is +/- infinity. So, for x values > that are a bit larger than 1 (or smaller than -1), I don't see how > this series converges to anything. For instance, if I did atan(2), > then (2)^n will grow a lot faster than n itself, so each term will be > larger than the last. > > Or am I missing something here? > > Thanks. > > Dave
Yes Dave, you are right. The above series converges only for x^2 <= 1.
"Table of Integrals and Products" by Gradshteyn and Ryzhik in Eq. 1.644 gives a series expansion that holds for all values of x^2 < infinity. (The book should be available in any good college library.) It is much too messy to try and reproduce here. And probably not a good way to compute the atan.