2. Let P E R. Consider the series sin(kx) KP k=1 (a) Prove that the series converges absolutely uniformly on R for p > 1

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2 Let P E R Consider The Series Sin Kx Kp K 1 A Prove That The Series Converges Absolutely Uniformly On R For P 1 1 (47.13 KiB) Viewed 36 times

2. Let P E R. Consider the series sin(kx) KP k=1 (a) Prove that the series converges absolutely uniformly on R for p > 1. (b) Using the fact that sin(kx) sin() = (cos((k-1) 1) - m € N, - Cos s((k+) z)), show that for any (m+1)x 771 sin (m) sin 2 Fm(x) = sin(kx) sin (5) k=1 71 sin(kx) (c) Let n € N. Define S₁ (2) => Show that for any n = N, k k=1 1 Sn(x) = = 12 + 1 F₁(x) + Σ Fk (2) ( = = - + Σ Fk ( x ) ( ² − k ² + 1). +1 k+1 k=1 (d) Use (b) and (c), or otherwise, prove that for any & such that 0 < d < T₂ converges k=1 uniformly on [6, 27 - 8]. sin(kx) k