Does the maximum error when applying Taylors series decrease as $n$ increases? (Lagrange Form of the Remainder) The title's kind of self explanatory

In the Lagrange form of the remainder I intuitively know (though I'm not sure I'm right) that as $n$ increases, the maximum error would decrease since $n$ implies how many terms from the Taylor polynomials you use. And apparently the more terms you use to define a function at a certain point you will get a more accurate value.

*

*Is my 'intuition' correct


*Any ways to prove that? My intuition says I'm right, but no matter how hard I stare at the Lagrange thing, I can't figure out why that is so.
 A: Marty Cohen’s answer is correct, but I think more exposition is valuable. Especially since Runge’s Phenomenon isn’t about Taylor expansions exactly.
Indeed though, we may consider the function $f(x) = \frac{1}{1 + x^2}$ Consider taking the Taylor expansion around $0$. This Taylor expansion is given by:
$$P_k(x) = \sum_{n=0}^k (-1)^k \cdot x^{2k}$$
Now this Taylor series has a radius of convergence of 1. Let’s look at what happens outside that radius.
For ease of argumentation, let’s look at what happens at $x=2$ on the even-degree Taylor approximations. We know that $f(2) = \frac{1}{5}$. However, we may show that:
$$\begin{align*} |P_{2k}(2)| = | 1 - 2^2 + 2^4 - 2^6 + \cdots + (-1)^{2k} 2^{4k}| = \frac{1 + 4^{2k+1}}{5} \end{align*}$$
This expression is clearly increasing in $k$. It easily follows that as $k \rightarrow \infty$, we have that $P_{2k}(2) \rightarrow \infty$. Similar analysis shows that the Taylor approximation goes to $-\infty$ On the odd-degree Taylor approximations.
If you want to play around, I’ve prepared a Desmos file for you: https://www.desmos.com/calculator/vloova1c1x
So in general, it is not true that the Taylor Remainder is decreasing.
When can we say for sure that the Taylor remainder is decreasing? Most notably, when the Taylor series is convergent to the function!
A: Nope. A well-known example is
$\dfrac1{1+x^2}$
with uniformly spaced interpolation points
in $[-1, 1]$.
This is known as
Runge's phenomenon.
See
https://en.wikipedia.org/wiki/Runge%27s_phenomenon
