18.100B.Homework5

18.100B.Homework5 - 18.100B/C: Fall 2008 Homework 5...

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Unformatted text preview: 18.100B/C: Fall 2008 Homework 5 Available Wednesday, October 8 Due Wednesday, October 15 Turn in the homework by 11am on Wednesday, October 15, in 2-108. For 18.100B it should be put in the bin corresponding to the lecture you regularly attend (regardless of which one you may be officially enrolled in). If you are enrolled in 18.100C put your homework in the C bin in any case. 1. (15pts) Assume that a positive term series ∞ an diverges. Study the behavior of the n=1 series ∞ ∞ an an a) and b) . 2 + n2 an 2 + an n=1 n=1 (Hint: for b) consider separately the cases where an does resp. does not converge to 0.) 2. • a) (15pts) Let n ≥ 1 and let a1 , a2 , ..., an and b1 , b2 , ..., bn be real numbers. Verify that 2 n 1 + 2 ai bi i=1 n n n 2 n a2 i (ai bj − aj bi ) = i=1 j =1 b2 j i=1 j =1 and conclude the Cauchy-Schwarz inequality 1 2 n n 1 2 n a2 i ai bi ≤ b2 j i=1 i=1 . j =1 Then use the Cauchy-Schwarz inequality inequality to prove the triangle inequality 1 2 n (ai + bi )2 1 2 n a2 i ≤ (Hint: square both sides!) • b) (10pts) Let now X = {(an ) ⊂ R / ∞ n=1 2 a2 < ∞} and define the norm n 1 2 a2 n = . n=1 Use part a) to show that (X, · 2) . j =1 ∞ (an ) b2 j + i=1 i=1 1 2 n is a normed space. 3. • a) (5pts) Prove that ∞ n=1 1 = 1. n(n + 1) (Hint: The partial sums can be written as telescoping sum (a1 − a2 ) + (a2 − a3 ) + . . . + (an−1 − an ) = a1 − an . ) • b) (10pts) Assume that an , bn > 0 for all n ≥ n0 and assume that an = L > 0. n→∞ bn lim Prove that ∞ n=1 an and ∞ n=1 bn either both converge or both diverge. 4. (b: 5pts, c: 10pts) Page 78 of Rudin, problem 6 part b) and part c). 5. (10pts each part) Page 79 of Rudin, problem 12. 6. (10pts) Page 81 of Rudin, problem 19. 2 ...
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This note was uploaded on 12/07/2011 for the course MATH 18.100B taught by Professor Prof.katrinwehrheim during the Fall '10 term at MIT.

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