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p12 - = x n 2 = ·· = x m = y:= 1 n n ∑ k =1 x k#4(a...

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Math 5615H: Introduction to Analysis I. Fall 2011 Homework #12 (due on Wednesday, November 30). 50 points are divided between 5 problems, 10 points each. #1. Let a j and b j,k be real numbers defined for all j, k = 0 , 1 , 2 , . . . , such that j =0 | a j | ≤ A = const < , | b j,n | ≤ B = const < for all j, n ; and lim n →∞ b j,n = 0 for each j . Show that lim n →∞ j =0 a j b j,n = 0. #2. Find the limit lim n →∞ ( 10 n 10 + 8 n 9 - n ) , using the facts that lim n →∞ e α n - 1 α n = 1 , lim n →∞ ln(1 + β n ) β n = 1 if α n 0 , β n 0 . #3. Let f ( x ) be a real-valued function on R , such that f ( x 1 + x 2 2 ) f ( x 1 ) + f ( x 2 ) 2 for all x 1 , x 2 R . (1) Show that f ( 1 n n k =1 x k ) 1 n n k =1 f ( x k ) for all natural n and x k R . (2) Hint. First prove (2) for n = 2 j , j = 1 , 2 , 3 , . . . , using induction. Then for an arbitrary natural n 3 and x 1 , x 2 , . . . , x n R , use (2) with m = 2 j > n instead of n , and x n +1 =
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Unformatted text preview: = x n +2 = ··· = x m = y := 1 n n ∑ k =1 x k . #4. (a). Show that the property (1) holds true for f ( x ) := e x . (b). Use the inequality (2) with f ( x ) := e x for the proof of the inequality ( a 1 a 2 ··· a n ) 1 /n ≤ 1 n ( a 1 + a 2 + ··· + a n ) for all natural n and a k ≥ . #5. Using the fact (Problem #4 in Homework #9) that s n := 1 + 1 2 + 1 3 + ··· + 1 n-ln n → γ = const as n → ∞ , evaluate the sum of the series ∞ ∑ n =1 (-1) n-1 n = 1-1 2 + 1 3- ··· . Hint. Express σ 2 n := 1-1 2 + 1 3- ··· + 1 2 n-1-1 2 n in terms of s 2 n and s n ....
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