1. **Stating the problem:** We want to simplify the expression $$\frac{1}{z} = \frac{1}{R + j\omega L} + j\omega C$$ and check the correctness of the working for the specific values $$R=1$$, $$L=2$$, and $$C=0.5$$. 2. **Given values:** $$R = 1, \quad L = 2, \quad C = 0.5$$ 3. **Substitute values:** $$\frac{1}{z} = \frac{1}{1 + j\omega 2} + j\omega 0.5$$ 4. **Find common denominator:** To add the two terms, write the second term with denominator $$1 + j\omega 2$$: $$\frac{1}{z} = \frac{1}{1 + j\omega 2} + \frac{j\omega 0.5 (1 + j\omega 2)}{1 + j\omega 2}$$ 5. **Expand numerator of second term:** $$j\omega 0.5 (1 + j\omega 2) = j\omega 0.5 + j^2 \omega^2 (0.5 \times 2) = j\omega 0.5 - \omega^2$$ 6. **Add numerators:** $$\frac{1}{z} = \frac{1 + j\omega 0.5 - \omega^2}{1 + j\omega 2}$$ 7. **Check the mistake:** The original working missed the initial "1" in the numerator when combining terms. The numerator should be $$1 + j\omega 0.5 - \omega^2$$, not just $$j^2 \omega^2 + j\omega 0.5$$. 8. **Summary:** The error was forgetting to add the "1" from the first fraction's numerator when combining the two terms over a common denominator. **Final simplified expression:** $$\frac{1}{z} = \frac{1 + j\omega 0.5 - \omega^2}{1 + j\omega 2}$$