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
$$
01, x_{101}>x_{3} \text {, so }
$$
$$
1<\frac{1}{x_{1}}-\frac{1}{x_{101}}<2 \text {, }
$$

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