Example 11 (2002-2003 Finnish High School Mathematics Competition) Let the sequence $\left\{x_{n}\right\}$ satisfy $x_{1}=\frac{1}{3}$ and $x_{k+1}=$ $x_{k}+x_{k}^{2}, k=1,2, \cdots$ Then the value of $\frac{1}{x_{1}+1}+\frac{1}{x_{2}+1}+\cdots+\frac{1}{x_{20 \times 2}+1}$ lies between which two consecutive integers?
Solving: By the recursive relation, it is easy to prove by induction that $01, x_{2003}>x_{3}>1$, so $2<\frac{1}{x_{1}}-\frac{1}{x_{2003}}<3$,
thus the value of $\frac{1}{x_{1}+1}+\frac{1}{x_{2}+1}+\cdots+\frac{1}{x_{2002}+1}$ lies between 2 and 3.