Example 7 Let $f(n)=\frac{1}{\sqrt[3]{n^{2}+2 n+1}+\sqrt[3]{n^{2}-1}+\sqrt[3]{n^{2}-2 n+1}}$, find the value of $f(1)+f(3)+f(5)+\cdots$ $+f(999997)+f(999999)$.
Analyzing this problem seems difficult at first, but as long as we notice that
$$
f(n)=\frac{\sqrt[3]{n+1}-\sqrt[3]{n-1}}{(\sqrt[3]{n+1})^{3}-(\sqrt[3]{n-1})^{3}}=\frac{1}{2}(\sqrt[3]{n+1}-\sqrt[3]{n-1})
$$

the problem is easily solved. Therefore,
$$
f(1)+f(3)+f(5)+\cdots+f(999997)+f(999999)=\cdots \cdots=\frac{1}{2}\left(\sqrt[3]{10^{6}}-\sqrt[3]{0}\right)=50
$$