8・13 Let the sequence $\left\{x_{n}\right\}$ satisfy: $x_{1}=\frac{1}{2}$, and
$$x_{k+1}=x_{k}+x_{k}^{2}, k=1,2, \cdots$$

Find the integer part of $\frac{1}{x_{1}+1}+\frac{1}{x_{2}+1}+\cdots+\frac{1}{x_{100}+1}$.
［Solution］From the recursive relation, it is easy to know
$$01, x_{101}>x_{3}$$ which implies
$$1<\frac{1}{x_{1}}-\frac{1}{x_{101}}<2$$

Thus, the integer part of $\frac{1}{x_{1}+1}+\frac{1}{x_{2}+1}+\cdots+\frac{1}{x_{100}+1}$ is 1.