(a) Write a program for calculating the A_m’s in Example 1 and extend the table to m = 15. Verify numerically that α_m ≈ (m - 1/4)π and compute the error for m = 1, · · ·, 10.

(b) Graph the initial deflection f(r) in Example 1 as well as the first three partial sums of the series. Comment on accuracy.

(c) Compute the radii of the nodal lines of u₂, u₃, u₄ when R = 1. How do these values compare to those of the nodes of the vibrating string of length 1? Can you establish any empirical laws by experimentation with further u_m?