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hw5 - x 4 16(2 Show that the equation x = 1 Z x cos t t 2 4...

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Math 135A, Winter 2008 Homework #5 (due 2/7/2008) Routine problems: § 11.4: #5, #7, #9, #12, #26, #27, #36 § 11.5: #3, #9, #21, #33, #34 § 11.6: #5, #9, #13, #19, #21, #24, #37 § 11.7: #7, #13,, #21, #38 To hand in: (1) Let { a n } be the sequence defined inductively by a 1 = 1, a n +1 = 1 a 4 n + 16 . (a) Show that { a n } is a Cauchy sequence. (b) Show that { a n } converges to a solution of the equation x 5 + 16 x - 1 = 0. (c) Show that if { b n } is the sequence defined by b 1 = 2, b n +1 = 1 b 4 n + 16 , then { b n } is convergent, and lim n →∞ b n = lim n →∞ a n . Hint: Consider the function f : R R defined by f ( x ) = 1
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Unformatted text preview: / ( x 4 + 16) . (2) Show that the equation x = 1 + Z x cos( t ) t 2 + 4 dt has one and only one solution. (3) Recall that (by definition) lim x →∞ f ( x ) = L if and only if for every real number ± > 0 there is a real number x such that | f ( x )-L | < ± for all x > x . Prove the following: lim x →∞ f ( x ) = L if and only if lim t → + f (1 /t ) = L. (4) Show that for any real number c , lim x →∞ ± x + c x-c ² x = e 2 c . 1...
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