4. As shown in Figure 2, construct a regular hexagon inside a circle with radius $r=1$, then construct the incircle of the hexagon, and again construct a regular hexagon inside this incircle, and so on, continuing infinitely. Let $S_{n}$ be the sum of the areas of the first $n$ circles. Given the positive number $\xi=3 \pi\left(\frac{3}{4}\right)^{99}$. If $\left|S_{n}-4 \pi\right|<\xi$, then the value of $n$ is ( ).
(A) all natural numbers greater than 100
(B) a finite number of natural numbers greater than 100
(C) all natural numbers not greater than 100
(D) a finite number of natural numbers not greater than 100
4. A.

Let the radius of the $n$-th circle be $r_{n}$.
It is easy to see that $r_{n}=\frac{\sqrt{3}}{2} r_{n-1}$, the area of the circle $a_{n}=\frac{3}{4} a_{n-1}$, $a_{1}=\pi r_{1}^{2}=\pi$.
Then $S_{n}=\frac{1-\left(\frac{3}{4}\right)^{n}}{1-\frac{3}{4}} \cdot \pi r_{1}^{2}=4 \pi\left[1-\left(\frac{3}{4}\right)^{n}\right]$.
By $\left|S_{n}-4 \pi\right|=4 \pi\left(\frac{3}{4}\right)^{n}100 \text {. }$