problemset3

Show that x2n1 x x2n 1 19 let a b 0 and let n

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Unformatted text preview: ; 0, and let n b e a positive integer. Show that xn 1 � . 2 + · · · + x2n 1+x+x 2n + 1 16. Let a, b > 0, a + b = 1, and q > 0. Show that � �q � �q 1 1 5q ∈ q−1 . a+ + b+ a b 2 17. Let x, y > 0 with x = y , and let m and n be positive integers. Show ≤ that xm y n + xn y m < xm+n + y m+n . 18. Let x > 0 but x = 1, and let n be a positive integer. Show that ≤ x2n−1 + x < x2n + 1. 19. Let a > b > 0, and let n be a positive integer greater than 1. Show that � � � n n n a − b < a − b. 20. Let a, b, x > 0 and a = b. Show that ≤ � �b+x � � a+x ax > . b+x b 21. Let a > b > 0, and let n be a positive integer greater than 1. Show that for k ∈ 0, � � n n an + k n − bn + k n � a − b. 22. Let x ∈ 0, and let m and n be real numbers such that m ∈ n > 0. Show that 1 − xm+n . (m + n)(1 + xm ) ∈ 2n 1 − xn � 23. Let ai ∈ 0 for 1 � i � n, and let n=1 ai = 1. Let 0 � xi � 1 for i 1 � i � n. Show that a1 a2 an 1 + +···+ � . a1 a2 1 + x1 1 + x2 1 + xn 1 + x 1 x2 · · · x an n 7 24. If a1 , . . . , an+1 are positive real numbers with a1 = an+1 , show that n n � � ai �n � ai+1 ∈ . ai+1 ai i=1 i=1 25. Let {a1 , a2 , . . . , an } and {b1 , b2 , . . . , bn } be two sets of real numbers with b1 ∈ b2 ∈ · · · ∈ bn ∈ 0. Put sk = a1 + a2 + · · · + ak for k = 1, 2, . . . , n; and let M and m denote respectively the largest and smallest of the numbers s1 , s2 , . . . , sn . Show that mb1 � n � i=1 a i bi � M b 1 . 26. Show that for any real numbers a1 , a2 , . . . , an , �n �2 n n � ai � � ai aj � . i i+j−1 i=1 i=1 j =1 27. Let f and g be real-valued functions defined on the set of real numbers. Show that there are numbers x and y such that 0 � x � 1, 0 � y � 1, and |xy − f (x) − g (x)| ∈ 1/4. 28. Let t > 0. Show that t� − � t � 1 − � , if 0 < � < 1 and t� − � t ∈ 1 − � , if � > 1. 29. Show that for a...
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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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