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problemset3 - 18.S34(FALL 2006 PROBLEMS ON INEQUALITIES 1...

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18.S34 (FALL 2006) PROBLEMS ON INEQUALITIES 1. Let a be a real number and n a positive integer, with a > 1. Show that n +1 n 1 n a 1 n a 2 a 2 . 2. Let x i > 0 for i = 1 , 2 , . . . , n . Show that 1 1 1 2 ( x 1 + x 2 + + x n ) + + + n . · · · x 1 x 2 · · · x n 3. If x i > 0 , q i > 0 for i = 1 , 2 , . . . , n , and q 1 + + q n = 1, show that · · · x q 1 x q n q 1 x 1 + + q n x n . 1 n · · · · · · 4. For p > 1 and a 1 , a 2 , . . . , a n positive, show that n n a 1 + a 2 + + a k p p p p · · · < a k . k p 1 k =1 k =1 5. If a n > 0 for n = 1 , 2 , . . . , show that n a 1 a 2 a n e a n , · · · n =1 n =1 provided that n =1 a n converges. 6. Let 0 < x < �/ 2. Show that 1 x sin x x 3 . 6 7. Show that 1 + 1 2 + 1 3 + · · · + 1 n > 2 n + 1 2 . 5
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8. Let a 1 a 2 a n , , . . . , b 1 b 2 b n be n fractions with b i > 0 for i = 1 , 2 . . . , n . Show that the fraction a 1 + a 2 + + a n · · · b 1 + b 2 + + b n · · · is contained between the largest and smallest of these n fractions. 9. For n = 1 , 2 , 3 , . . . let 1000 n x n = . n ! Find the largest term of the sequence. 10. Suppose that a 1 , a 2 , . . . , a n with n 2 are real numbers larger than 1, and moreover all a j ’s have the same sign. Show that (1 + a 1 )(1 + a 2 ) (1 + a n ) > 1 + a 1 + a 2 + · · · · · · + a n . 11. Show that 1 3 5 2 n 1 1 2 · 4 · 6 · · · 2 n < 2 n + 1 . 12. Prove Chebyshev’s inequality : If a 1 and b 1 a 2 · · · a n b 2 · · · b n , then n n n 1 1 1 n a k n b k n a k b k . k =1 k =1 k =1 Generalize to more than two sets of increasing sequences. 13. Let n be a positive integer larger than 1, and let a > 0. Show that n 1 + a + a 2 + + a n + 1 . · · · a + a 2 + a 3 + + a n 1 n 1 · · · 14. Show that if a > b > 0, then A < B , where 1 + a + + a n 1 1 + b + + b n 1 A = · · · , B = · · · . 1 + a + + a n 1 + b + + b n · · · · · · 6
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15. Let x > 0, and let n be a positive integer. Show that x n 1 . 1 + x + x 2 + + x 2 n 2 n + 1 · · · 16. Let a, b > 0 , a + b = 1, and q > 0. Show that q q 1 1 5 q a + + b + .
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