smallest integer value of $n$ such that $|S_n-S_\infty|<0.001$ 
If $S_n$ and $S_\infty$ are sums to $n$ terms and sum to infinity of a geometric progression $3,-\frac{3}{2},\frac{3}{4},...$ respectively, find the smallest integer value of $n$ such that $|S_n-S_\infty|<0.001$

My attempt,
$$|S_n-S_\infty|<0.001$$
$$|\frac{3[1-(-\frac{1}{2})^n]}{1-(-\frac{1}{2})}-\frac{3}{1-(-\frac{1}{2})}|<0.001$$
$$|2[1-(-\frac{1}2)^n]-2|<0.001$$
How to proceed? Thanks in advance.

$$|2[1-(-\frac{1}2)^n]-2|<0.001\\
|-2(\frac{-1}2)^n|<0.001\\
+2|(\frac{-1}2)^n|<0.001\\\to \text{abs function properties  } |\frac{-1}{2^n}|=\frac{1}{2^n}\\
+2.\frac{1}{2^n}<\frac{1}{1000}\\
\frac{2^n}{2}>1000\\
2^{n-1}>1000\\\text{note that  } 2^{10}=\color{red} {1024>1000}\\\to \\n-1\geq 10\\n \geq 11$$
