Prove Theorem 13.8 by filling in the details of the

Chapter 13, Problem 13.37

(choose chapter or problem)

Prove Theorem 13.8 by filling in the details of the following argument. Denote the Fourier coefficients of f (x) by lower case letters, and those of f (x) by upper case. Show that A0 = 0, An = n L bn , and Bn = n L an . Show that 0 A2 n 2 n |An | + 1 n2 for n = 1, 2, , with a similar inequality for Bn . Add these two inequalities to obtain 1 n (|An |+|Bn |) 1 2 (A2 n + B2 n ) + 1 n2 . Hence show that |an |+|bn | L 2 (A2 n + B2 n ) + L (n2) . Thus show by comparison that n=1 (|an |+|bn |) converges. Finally, show that |an cos(nx/L) + bn sin(nx/L)||an |+|bn | and apply a theorem of Weierstrass on uniform convergence.