1. **Problem:** Simplify the expression $1 - \frac{1}{2} + \frac{1}{4} - \frac{1}{8} + \frac{1}{16} - \frac{1}{32}$. 2. **Formula and rules:** This is a finite geometric series with first term $a = 1$ and common ratio $r = -\frac{1}{2}$. 3. **Sum of geometric series:** The sum of $n$ terms is given by $$S_n = a \frac{1 - r^n}{1 - r}$$ 4. **Calculate:** Here, $n=6$, so $$S_6 = 1 \times \frac{1 - \left(-\frac{1}{2}\right)^6}{1 - \left(-\frac{1}{2}\right)} = \frac{1 - \frac{1}{64}}{1 + \frac{1}{2}} = \frac{\frac{63}{64}}{\frac{3}{2}}$$ 5. **Simplify:** $$\frac{\frac{63}{64}}{\frac{3}{2}} = \frac{63}{64} \times \frac{2}{3} = \frac{63 \times 2}{64 \times 3} = \frac{126}{192}$$ 6. **Reduce fraction:** $$\frac{126}{192} = \frac{\cancel{126}^{63}}{\cancel{192}^{96}} \times \frac{2}{3} = \frac{21}{32}$$ 7. **Final answer:** $$\boxed{\frac{21}{32}}$$