problemset3

Let aj a1 a2 aj and bj b1 b2 bj show

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Unformatted text preview: ny real number x and any positive integer n we have � � �n � � �� sin kx � � � 2 �. � k� k =1 30. Show that if x is larger than any of the numbers a1 , a2 , . . . , an , then 1 1 1 n + +···+ ∈ . 1 x − a1 x − a2 x − an x − n (a1 + a2 + · · · + an ) 8 31. Show that �� � �� � �� � � n n n + +···+ � n(2n − 1). 1 2 n 32. Let y = f (x) b e a continuous, strictly increasing function of x for x ∈ 0, with f (0) = 0, and let f −1 denote the inverse function to f . If a and b are nonnegative constants, then show that �a �b ab � f (x)dx + f −1 (y )dy. 0 0 33. Show that for t ∈ 1 and s ∈ 0, ts � t log t − t + es . 34. Let a1 /b1 , a2 /b2 , . . ., with each bi > 0, be a strictly increasing sequence. Let Aj = a1 + a2 + · · · + aj , and Bj = b1 + b2 + · · · + bj . Show that the sequence A1 /B1 , A2 /B2 , . . . is also strictly increasing. 35. Let m, n be positive integers, and let a1 , a2 , . . . , an be positive real numbers. For i = 1, 2, 3 . . . put an+i = ai and bi = ai+1 + ai+2 + · · · + ai+m . Show that mn a 1 a 2 · · · a n < b 1 b 2 · · · b n , except if all the ai are equal. 36. Let a1 , a2 , . . . , an be real numbers. Show that � � min (ai − aj )2 � M 2 a2 + · · · + a2 , 1 n ai =aj � where M2 = 9 12 . n(n2 − 1) 37. Let x and a be real numbers, and let n be a nonnegative integer. Show that � �(n+1)/2 |x − a|n |x + na| � x2 + na2 . 38. Given an arbitrary finite set of n pairs of positive real numbers {(ai , bi ) : i = 1, 2, . . . , n}, show that �n � n n � � � (xai + (1 − x)bi ) � max ai , bi , i=1 i=1 i=1 for all x � [0, 1]. Equality is attained only at x = 0 or x = 1, and then if and only if �n �� n � � ai − b i � ai − b i ∈ 0. ai bi i=1 i=1 39. Show that if m and n are positive integers, then the smallest of the � � � numbers n m and m n cannot exceed 3 3. 40. Show that if a ∈ 2 and x > 0, then ax + a1/x � ax+1/x , with equa...
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This note was uploaded on 01/22/2014 for the course MATH 18.S34 taught by Professor Hartleyrogers during the Fall '07 term at MIT.

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