# s8 - Math 5615H Introduction to Analysis I Homework#8...

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Math 5615H: Introduction to Analysis I. Fall 2011 Homework #8. Problems and Solutions. #1. Let 0 < x 1 = a < x 2 = b be arbitrary real number, and let x n := 1 2 ( x n - 2 + x n - 1 ) for n = 3 , 4 , 5 ,.... Show that the sequence { x n } converges, and ﬁnd its limit. Solution. The sequence y n := x n +1 - x n satisﬁes y 1 = b - a , and y n +1 = x n +2 - x n +1 = 1 2 ( x n + x n +1 ) - x n +1 = - 1 2 · y n . By iteration, we get y n +1 = q n y 1 , where q := - 1 / 2. Then x n +2 = x 1 + ( x 2 - x 1 ) + ( x 3 - x 2 ) + ··· + ( x n +2 - x n +1 ) = a + y 1 + y 2 + ··· + y n +1 = a + (1 + q + ··· + q n ) · y 1 = a + 1 - q n +1 1 - q · y 1 . Since q n = ( - 1 / 2) n 0 as n → ∞ , we get lim n →∞ x n = lim n →∞ x n +2 = a + y 1 1 - q = a + 2 3 · ( b - a ) = a + 2 b 3 . #2. Find the upper and lower limits of the sequence { s n } deﬁned by s 0 = 0; s 2 m = s 2 m - 1 2 ; s 2 m +1 = 1 2 + s 2 m . Solution.

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s8 - Math 5615H Introduction to Analysis I Homework#8...

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