1. Given the sets $M=\{x, x y, \lg (x y)\}$ and $N=\{0,|x|, y\}$, and $M=N$. Then, $\left(x+\frac{1}{y}\right)+\left(x^{2}+\frac{1}{y^{2}}\right)+\left(x^{3}+\right.$ $\left.\frac{1}{y^{3}}\right)+\cdots+\left(x^{2001}+\frac{1}{y^{2001}}\right)$ is equal to
Given $\because M=N$,
$\therefore 0 \in M$, and $x, y$ cannot both be 0, then
$\lg (x y)=0, x y=1$.
Thus, $1 \in N$. If $y=1$, then $x=1$, which contradicts the distinctness of the elements in the set. Therefore, $|x|=1$, and $x=-1$, so $y=-1$.

Therefore, $x+\frac{1}{y}=-2 \cdot x^{2}+\frac{1}{y^{2}}=2$, $x^{3}+\frac{1}{y^{3}}=-2, \cdots, x^{2000}+\frac{1}{y^{2000}}=2$, $x^{2001}+\frac{1}{y^{3} \frac{1}{301}}=-2$, and the sum is -2.