Answer Happy

Please do Part c and d!
sin(kx) 2. Let p E R. Consider the series kP k=1 (a) Prove that the series converges absolutely uniformly on R for p > 1. (b) Using the fact that sin(kx) sin (1) * (cos((k - ) 2) - cos ((k + ) 2)), show that for any m EN sin (12) sin Fm(1) sin(kx) = sin (1) m (m+1). 2 k=1 (c) Let ne N. Define Sn(1) sin(kx) k Show that for any neN, k=1 1 1 Su(x) = x + F(x) + F(m) ( ) (1) +ΣFa () Sn n+1 k k + k=1 (d) Use (b) and (c), or otherwise, prove that for any ô such that 0) < 8 <a, uniformly on (8, 21 – 8]. sin(kx) k converges k=1 -