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p10 - ∑ a n if a a n = √ n 1 − √ n b a n = √ n 1...

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Math 5615H: Introduction to Analysis I. Fall 2011 Homework #10 (due on Wednesday, November 16). 50 points are divided between 4 problems. #1. (12 points). The sequence { a n } is deﬁned by a 1 = 0 , and a n +1 = 2 a n / 2 for n = 1 , 2 , 3 , . . . . Prove that { a n } is convergent and ﬁnd its limit. You can use without proof the fact that the function y = f ( x ) = a x , where a > 0 , a ̸ = 1, is strictly convex , i.e. f ( tx 1 + (1 t ) x 2 ) < t f ( x 1 ) + (1 t ) f ( x 2 ) for all x 1 < x 2 and 0 < t < 1 . #2. (12 points, Exercise 6(a–c) on p.78). Investigate the behavior (convergence or divergence) of
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Unformatted text preview: ∑ a n if ( a ) a n = √ n + 1 − √ n ; ( b ) a n = √ n + 1 − √ n n ; ( c ) a n = ( n √ n − 1 ) n . #3. (10 points). Using the fact that ∞ ± n =1 1 n 2 = π 2 6 , ﬁnd ∞ ± n =1 ( − 1) n-1 n 2 . #4. (16 points, Exercise 9(a–d) on p.79). Find the radius of convergence of each of the following power series: ( a ) ± n 2 z n , ( b ) ± 2 n n ! z n , ( c ) ± 2 n n 2 z n , ( d ) ± n 3 3 n z n ....
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