Homework_Set_5

Homework_Set_5 - Math 1a Fall 2005 HOMEWORK SET 5 DUE AT...

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Unformatted text preview: Math 1a Fall 2005 HOMEWORK SET 5 DUE AT NOON ON MONDAY, OCTOBER 31, 2005 (1) [50 points] Define for n = 0, 1, 2, . . . , π In = 0 sinn x dx (a) Prove that In+2 = (n + 1)In − (n + 1)In+2 (Hint: Integrate by parts and use sin2 x + cos2 x = 1.) (b) Using (a), prove that lim n→∞ In+2 =1 In (c) Prove that In+1 ≤ In and, using (b), then prove lim n→∞ In+1 =1 In (d) Write I2n in terms of I0 = π and I2n+1 in terms of I1 = 2. (e) Prove Wallis’ formula: lim n→∞ 2·2 3·1 4·4 3·5 6·6 5·7 ··· 2n · 2n (2n + 1)(2n − 1) = π 2 (f) Use Mathematica or Maple to find the 2 · LHS of (e) for n = 1 and 10 and 100 and compare to π . (2) [25 points] (a) Apostol, page 180, problem 22. (b) Apostol, page 181, problem 27 (3) [25 points] Apostol, page 187, problem 9. 1 ...
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This note was uploaded on 03/30/2010 for the course MA 1a taught by Professor Borodin,a during the Fall '08 term at Caltech.

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