Why $\tan x$ is not infinity at all the points > $\frac{\pi}{2}$

Question

Looking at the series expansion of $\tan x$, it appears as the monotonically increasing function which we know isn't true. So, how can we describe it's periodicity from it's Maclaurin series?

Answer 1

I believe that it's because the function is discontinuous (and undifferentiable) at $x= \pm \frac{ \pi}{2}$. Since Taylor series rely only on continuity and looks at the derivatives at a single point, in theory it should rapidly diverge to infinity after $x= \pm \frac{ \pi}{2}$ (not sure if it actually does).

Answer 2

The Taylor series of a smooth function at a point need not converge everywhere; even if it does, it may not equal the function everywhere! In this case, the Taylor series at $0$ converges and agrees with tangent on the connected component of the graph containing the origin; but this isn't the whole graph! So the rest of the tangent function is invisible to the Taylor series at zero.