- Sin Kr 2 Let Pe R Consider The Series A Prove That The Series Converges Absolutely Uniformly On R For P 1 B Us 1 (36.21 KiB) Viewed 20 times
sin(kr) 2. Let pe R. Consider the series (a) Prove that the series converges absolutely uniformly on R for p > 1. (b) Using the fact that sin(kx) sin() = 1 (cos((k - ) ) (k+ 1) )), show that for any meN - COS m sin ("04) sin((m+1)2 2 Fm(x) = sin(kx) sin () k=1 72 (c) Let ne N. Define Sn(t)= sin(kx) k Show that for any n EN, k=1 1 1 Sn(x) ntifu() + F() (xtu). () ΣΕ( n+1 k +1 kel 00 sin(k.c) k (d) Use (b) and (c), or otherwise, prove that for any & such that 0 <d<1, uniformly on (8,21 - 8). converges k