Example 3 Find the value of $\sum_{k=0}^{997}(-1)^{k} C_{1998}^{k}$.

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Example 3 Find the value of $\sum_{k=0}^{997}(-1)^{k} C_{1998}^{k}$.
Solution: $\begin{array}{l} \sum_{k=0}^{997}(-1)^{k} C_{1998}^{k} \\ =1-\left(C_{1997}^{0}+C_{1997}^{1}\right)+\left(C_{1997}^{1}+C_{1997}^{2}\right)-\left(C_{1997}^{2}+C_{1997}^{3}\right) \\ +\cdots+\left(C_{1997}^{995}+C_{1997}^{996}\right)-\left(C_{1997}^{996}+C_{1997}^{997}\right)=-C_{1997}^{997} .\end{array}$