I'm reading the book, functions of one complex variable, John B. Conway. in chapter 3.2, page 38, it is written that each of the series $\sin z$ and $\cos z$ has an infinite radius of convergence, and so they are analytic in $C$.
I can't understand why the infinite radius of convergence results in that they are analytic.
Can anyone help?
You should use this terminology (you can use different words but the point is to use those concepts) :
Analytic : represented by a power series on some disk around each point,
Entire : represented by a single power series on the whole complex plane (the Taylor series at $z=0$ or at any point)
Holomorphic : complex differentiable on some disk around each point.
The main theorem is the Cauchy integral theorem/formula which proves that holomorphic on $\Bbb{C}$ implies analytic on $\Bbb{C}$ and entire.
The same works for disks : holomorphic on a disk implies analytic and represented by a single power series (the Taylor series at the center of the disk).
The converse is obvious : entire implies analytic and holomorphic.