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I am given this question:

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The question explicitly mentioned that I could assume that $R>>1/wC$. However, after this assumption, the energy dissipated should be 0 and, hence, $Q$ should tend to infinity. Is there something wrong with this question?

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I believe you need to take approximation only after separating the real and imaginary parts of the effective impedance. Do not use the approximation for the energy dissipation (real part) but use it for the imaginary part. This is the impedance I got:

$$Z=\frac{R}{1+\omega^2C^2R^2}+j(\omega L-\frac{\omega CR^2}{1+\omega^2C^2R^2})$$

and use the approximation only in the 2nd term so you get:

$$j(\omega L-\frac{1}{\omega C})$$

Probably you can further simplify the 1st term to

$$\frac{1}{\omega^2C^2R}$$

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