Example 8 Calculate
$$
\begin{array}{l}
1 \frac{1}{2048}+2 \frac{1}{1024}+4 \frac{1}{512}+\cdots+512 \frac{1}{4}+ \\
1024 \frac{1}{2} .
\end{array}
$$
Explanation: Since each number added to itself yields the next number, we can add $1 \frac{1}{2048}$. Therefore, the original expression is:
$$
\begin{aligned}
= & 1 \frac{1}{2048}+1 \frac{1}{2048}+2 \frac{1}{1024}+4 \frac{1}{512}+ \\
& \cdots+512 \frac{1}{4}+1024 \frac{1}{2}-1 \frac{1}{2048} \\
= & 2 \frac{1}{1024}+2 \frac{1}{1024}+4 \frac{1}{512}+\cdots+ \\
& 1024 \frac{1}{2}-1 \frac{1}{2048} \\
= & 4 \frac{1}{512}+4 \frac{1}{512}+\cdots+1024 \frac{1}{2}-1 \frac{1}{2048} \\
= & \cdots \cdots \\
= & 1024 \frac{1}{2}+1024 \frac{1}{2}-1 \frac{1}{2048} \\
= & 2049-1 \frac{1}{2048}=2047 \frac{2047}{2048} .
\end{aligned}
$$