TITLE: How to check the real analyticity of a function?
QUESTION [20 upvotes]: I recently learnt Taylor series in my class. I would like to know how is to possible to distinguish whether a function is real-analytic or not. First thing to check is if it is smooth. But how can I know whether the taylor series converges to the function? 
For example: $f(x)=\frac{1}{1-x}, x\in(0,1)$ has $n^{th}$ degree taylor polynomial $\sum_{k=0}^n x^k$. In this case, I understand that $f$ is analytic in its domain since the geometric series $\sum_{k=0}^\infty x^k$ for $x\in(0,1)$ converges to $\frac{1}{1-x}$. 
In general, what is the trick? For example, how to know if $\sin(x),\cos(x)$ are analytic?

REPLY [3 votes]: I agree with Betty Mock's thesis that complex analytic functions are usually easier to deal with than real analytic functions, but I don't think that entire functions (e.g. $e^z$, $\cos z$, $\sin z$) are a good example of that: the real-analytic story is the same as the complex-analytic story.
Looking around, I found this previous math.SE question which asks why if a real function $f$ is analytic at $a \in \mathbb{R}$ and $f(a) \neq 0$, then $\frac{1}{f}$ is analytic at $a$.  As the answers indicate, if you use complex analysis then it's just a matter of adapting the differentiability of the reciprocal to functions of a complex variable (no problem).  However, if you insist on showing directly that the Taylor series of $\frac{1}{f}$ has a positive radius of convergence at $a$....then this really is a pain, as several of the answers (including one due to me, where I reference a book on real analytic function theory but don't have the energy to reproduce the details) attest.
Already understanding why $f(x) = \frac{1}{x^2+1}$ is real analytic and that the radius of convergence of the Taylor series expansion at $a \in \mathbb{R}$ is precisely $\sqrt{1+a^2}$ is a quick pure-thought argument if you know the rudiments of complex analysis.  But trying to show this formula for the radius of convergence directly from the Taylor series expansion...there will be some actual work there, it seems to me.