Why is the domain of arctan power series |x|<=1?

[linyen416]

Why is the domain of arctan power series |x|<=1?
i understand why a power series has the domain |x|<1, but why is the power series of arctan |x|<=1?
thanks!

also, why are domains of the taylor expansions of coshx and sinhx: for all x?

 

[arildno]

What do you know about the complex number plane?

 

[mathwonk]

are you kidding? arctan is the integral of 1/(1+x^2). what makes it look undefined at some points of |z| = 1?