MAT 128B Study Guide - Midterm Guide: Bounded Function
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Xn=1 x2n: compute its radius r of convergence, does it converge at x = r, explain your answer. Xn=1 has radius of convergence that is 1. Hence the series converges for |y| < 1. 3 : at x = r = q 1. 0 x [0, 1) x = 1: the convergence cannot be uniform on [0, 1] because the pointwise limit f (x) is not continuous on [0, 1]. c. sup. |xn| = an which goes to zero as n . Hence the convergence is uniform on [0, a]. 3. (20 pts) show that p n=1 sin nx n3 converges to a continuous function on. We use the weierstrass m test. and sin nx n3. Hence the series converges uniformly on ( , ) and so the limiting function is continuous. 4. (20 pts) suppose fn(x) converges uniformly on [a, b] to a bounded function f (x) and gn(x) converges uniformly on [a, b] to a bounded function g(x).
