0 as n increases, which means that even the worst discrepancy between u and Un goes to zero as n gets large.

Multiplying out the terms, we have Q(vn) - 2(vn, Kv) + (v, Kv) -+ 0. (2) Since Vn converges to v componentwise, (vn, Kv) -+ (v, Kv). vll ? 1. vll :::; 1 since vis the weak limit of a sequence of unit vectors. vll = 1, and we have Q(v) = k. (5) Thus, v maximizes the Rayleigh quotient. i = (ui,

B) What is the radius of convergence in the complex c-plane of the Neumann series solution of the equation? u(x) = f(x) {21r + c Jo cos 2 (x- y)u(y)dy. 5. (a) Find the resolvent kernel for the following equation. (b) Is there a solution when c = 2 and f(x) = 1? u(x) = f(x) +c 1 1 [sin 11'(X- yWu(y)dy. 6. 6 Real Symmetric Matrices The number of linearly independent eigenvectors of an N x N matrix is at least as large as the number of distinct roots of the secular equation det(K- kl) = 0. Although the equation has N roots, if repeated roots are counted according to their multiplicity, the number of distinct roots 52 3.