Does |Taylor Series of $f$ - $f$| Converge Monotonically to $0$? Suppose that $T_n(x)$ be the sum of the first $n$ terms of the Taylor series of $f$ centered at $a$, and $\lim_{n\to \infty} T_n(b)=f(b)$. 
Is the difference $|T_n(b)-f(b)|$ decrease monotonically? 
If yes, why? Otherwise, can you give a counterexample?
Thanks.
 A: Nope it doesn't have to converge monotone, look at $f(x)=\sin(x)$. We center at $a=0$. Then $T_0=0$ and $T_0 (\pi \cdot k)=f(\pi \cdot k)=0$ with $k\in \mathbb{Z}$. But for sure  $T_1(\pi)=\pi\neq 0 =f(\pi)$. Furthermore choosing $k$ big enough you see that it will go farer apart in the first steps.
A: There is absolutely no reason to believe that adding a term to a Taylor series always improves the estimate it gives for a value $f(b)$ at a point distant from $a$ where the Taylor series is centred. A first approximation might be lucky and hit $f(b)$ on the nose (or be very close) for no particular reason; for instance the order$~0$ expansion of $f(x)=(x-1)x(x+1)$ at $0$ gives the right value for $b=-1$ and for $b=1$ (but not for any other $b\neq0$); adding the order$~1$ term certainly makes this perfect estimate worse.
You should also be aware that there is no reason that the Taylor series evaluated at $b$ should converge at all, or if it does that it should converge to $f(b)$. (You put those conditions in the hypotheses, which counterexamples must therefore respect, but they don't make any difference for the conclusion you seek). That we are tempted to expect such convergence is due to our over-exposition to pathologically well-behaved (namely analytic) functions.
