[Math] Why Taylor series does not converge for all x in the domain of the function

Question

Example:
$$
f(x)=\frac{1}{1+x} \qquad x\neq-1
$$
$$
f(x)=1-x+x^2-x^3+x^4-x^5+\;… \qquad |x| < 1
$$

Why Taylor series does not converge for all x in the domain of the function?

Answer 1

It's actually surprising that the Taylor series converges to the function anywhere other than $x=0$ -- functions that are equal to their Taylor series are actually rather rare.

Of course, most of the functions people actually study do have this property of being "analytic", so it's easy not to get a sense of how special this property is.

It turns out there is a pleasing geometric description of what the radius of convergence of a Taylor series will be: if you look at the complex plane, your analytic function may have some singularities. The Taylor series around any point will converge in a disk around that point whose radius is as large as it can possibly be without including any of those singularities.

Your particular $f$ has a singularity at $x = -1$, and that is its only singularity in the complex plane. Thus, the Taylor series around $x=0$ will have a radius of convergence equal to $1$. Its Taylor series around $ x = 41$ will have a radius of converge equal to $42$. Its Taylor series around $x = \mathbf{i}$ will have radius of convergence $\sqrt{2}$, and so forth.