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Math · Advanced Math
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1. In your last homework assignment, you showed that arctan(a) whenever -1 r 1. In this exercise, you will show that the identity remains valid when T -1.1 (a) Show that the series converges uniformly on the closed interval [0, 1]. Hint: First explain why the series converges for all r E 0,1 Letting s (z) be 1]. 2k +1 the sum of the series, and s (r) show that d(sn, s) On 2n+3 A 0,1. It might help to look back in your notes to our discussion of alternating Series (b) Explain why the result of (a) implies that the function s (T) defined by S (r) is continuous on 0, 1 (c) Deduce that holds when r 1. Hint: Use the fact that arctan (a) is continuous and that is already known to hold when r 1.

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