2. Let pe R. Consider the series sin(kr) kp (a) Prove that the series converges absolutely uniformly on R for p > 1. (b)

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2 Let Pe R Consider The Series Sin Kr Kp A Prove That The Series Converges Absolutely Uniformly On R For P 1 B 1 (24.77 KiB) Viewed 34 times

2. Let pe R. Consider the series sin(kr) kp (a) Prove that the series converges absolutely uniformly on R for p > 1. (b) Using the fact that sin(kx) sin () = 1 (cos((k - })) - - cos (k + ))), show that for any MEN sin (2) sin F.n()=sin(kI) 771 (m+1) 2 sin () k=1 (c) Let n € N. Define S (x) = sin(ka) Show that for any neN. k k=1 1 S.(a) = n*iFn(s) + F(-) ( x+i). = () +Σ F(α) 1- n +1 k=1 sin(kx) converges (d) Use (b) and (c), or otherwise, prove that for any & such that 0 < 8<, uniformly on (8,27 - 8). k=1