solutions for c

# solutions for c - Math 116 CALCULUS II SOLUTION FOR...

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Math 116 CALCULUS II SOLUTION FOR PRACTICE EXAM – FINAL C (12/05) December 5 (Thu), 2013 Instructor: Yasuyuki Kachi Line #: 23590. [I] (40pts) arctan x = 1 1 3 x 3 + 1 5 x 5 1 7 x 7 + 1 9 x 9 1 11 x 11 + ··· = s n = 0 p 1 P n 2 n + 1 x 2 n +1 . The radius of convergence R = 1. (2a) d dx ± arctan x ² = 1 x 2 + 1 . (2b) i + x =0 1 x 2 + 1 dx = b arctan x B + x =0 = arctan p + P arctan 0 = π 2 0 = π 2 . 1

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[II] (40pts) (1) Machin’s formula is as follows: π = s n = 0 p 1 P n 2 n + 1 16 5 2 n +1 4 239 2 n +1 = ± 16 5 1 4 239 1 ² 1 3 ± 16 5 3 4 239 3 ² + 1 5 ± 16 5 5 4 239 5 ² 1 7 ± 16 5 7 4 239 7 ² + 1 9 ± 16 5 9 4 239 9 ² 1 11 ± 16 5 11 4 239 11 ² + ··· (2) BBP formula is as follows: π = s n = 0 ± ² n 1 16 4 8 n + 1 2 8 n + 4 1 8 n + 5 1 8 n + 6 = ± 4 1 2 4 1 5 1 6 ² + 1 16 ± 4 9 2 12 1 13 1 14 ² + 1 16 2 ± 4 17 2 20 1 21 1 22 ² + 1 16 3 ± 4 25 2 28 1 29 1 30 ² + 1 16 4 ± 4 33 2 36 1 37 1 38 ² + .
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