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sin(kx) 2. Let P E R. Consider the series (a) Prove that the series converges absolutely uniformly on IR for p > 1. (b) Using the fact that sin(kx) sin () = (cos((k − ¹) x) - m EN, - COS m 2 Fm(x) = sin(kx) sin (2) sin((m+1) z sin (2) k=1 72 sin(kx) (c) Let n = N. Define S₂(x) = Σ Show that for any n E N, k k=1 n 1 Sn(x) 7 F₂(x) + [ Fx (x) ( 1/2 - (1-1). n+1 k+ k=1 (d) Use (b) and (c), or otherwise, prove that for any & such that 0 < d < π, Σ k=1 uniformly on [8, 2π – 8]. ((k+¹)x)), show that for any converges sin(kx)