This problem concerns a collection of N identical harmonic oscillators (perhaps an Einstein solid or the internal vibrations of gas molecules) at temperature T. The allowed energies of each oscillator are 0, hf, 2hf, and so on. We will use the infinite series 1 + x + x 2 + x 3 + · · · = 1 1 − x , 0 < x < 1 (a) Evaluate the partition function for a single harmonic oscillator. (b) Find an expression for the average energy of a single oscillator at temperature T. (c) What is the total energy of the system of N oscillators at temperature T? (d) Compute the heat capacity C = dU dT , and find the limit of C at T → 0: prove that it satisfies the third law of thermodynamics