I am doing a beginner's course in real analysis, so I am fairly new to it.
I've been told that a smooth function is infinitely differentiable, that is, all its derivatives exist. Moreover, a smooth function is called analytic at a point $a$ if the Taylor series of the function converges to the function in some neighbourhood of $a$.
My problem is, that these are merely definitions. I am not able to understand how I can show whether or not a given function is analytic at a given point.
$ln(1+x)$ for example, at $x = 0$. I don't see a method to check the convergence of that infinite series, other than the ratio test - which assures convergence in $(-1,1)$. So is the function analytic only in that interval? Is it possible that the Taylor series converges in a particular neighbourhood, but not to the value of the function at that point? Please help me prove or disprove this, and also let me know the ways to determine if a function is analytic or not, in general! Thanks :)