14. For $i=1,2, \cdots, n$, we have $\left|x_{i}\right|<1$, and $\left|x_{1}\right|+\left|x_{2}\right|+\cdots+\left|x_{n}\right|=2005+\mid x_{1}+$ $x_{2}+\cdots+x_{n}$. Find the minimum value of the positive integer $n$.
14. $n_{\min }=2006$ From the given, we have $2005=\left|x_{1}\right|+\left|x_{2}\right|+\cdots+\left|x_{n}\right|-\left|x_{1}+x_{2}+\cdots+x_{n}\right| \leqslant\left|x_{1}\right|+$ $\left|x_{2}\right|+\cdots+\left|x_{n}\right|<n$. Therefore, $n \geqslant 2006$.
On the other hand, let $x_{i}=\left\{\begin{array}{l}\frac{2005}{2006}, i \text { is odd; } \\ -\frac{2005}{2006}, i \text { is even. }\end{array}\right.$ They satisfy the conditions of the problem.