3. The sequence $\lg 1000, \lg \left(1000 \cdot \cos \frac{\pi}{3}\right)$, $\lg \left(1000 \cdot \cos ^{2} \frac{\pi}{3}\right), \cdots, \lg \left(1000 \cdot \cos ^{n-1} \frac{\pi}{3}\right)$, $\cdots$, when the sum of the first $n$ terms is maximized, the value of $n$ is ( ).
(A) 9
(B) 10
(C) 11
(D) Does not exist
3. (B).

Given $a_{n}=\lg \left(1000 \cdot \cos ^{n-1} \frac{\pi}{3}\right)=\lg \frac{1000}{2^{n-1}}$, we know that $\left\{a_{n}\right\}$ is a decreasing sequence.
Also, since $2^{9}=512,2^{10}=1024$, it follows that $a_{10}>0, a_{11}<0$, thus $S_{10}$ is the maximum.