- 8 2 Let P E R Consider The Series Sin Kx Kp K 1 Cos A Prove That The Series Converges Absolutely Uniformly On R 1 (57.64 KiB) Viewed 22 times
8 2. Let p E R. Consider the series sin(kx) kp k=1 = - COS (a) Prove that the series converges absolutely uniformly on R for p > 1. (b) Using the fact that sin(kx) sin (1) = (cos ((k – Ż) x) – ((k + ) x)), show that for any m EN sin (2) sin Fm(x) = sin(kx) m (m+1). 2 sin (3) k=1 n (c) Let n € N. Define Sn(x) = sin(kx) k k=1 Show that for any n EN, 1 1 Su(x) = m+īfu(2) + F(-) (4 x+1) Sn(2) ) + () +1 k k k=1 sin(kx) k converges (d) Use (b) and (c), or otherwise, prove that for any d such that 0 < d <a, uniformly on (8, 21 – 6). k=1