hw6 - lim n →∞ ± a √ n a 1 √ n 1 a 2 √ n 2 ··...

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Homework 6 – Math 118A, Fall 2009 Due on Thursday, November 19th, 2009 1. Consider the following space of sequences of real numbers: l p = ( { s n } ⊂ R ± ± ± ± X n =0 | s n | p < ) , 1 p < , and l = ² { s n } ⊂ R ± ± ± ± sup n 0 | s n | < ³ . For 1 p < , de±ne in l p the function: k s k p = ( X n =0 | s n | p ) 1 /p , and in l : k s k = sup n 0 | s n | . (a) Prove that the given functions are norms in the corresponding spaces. (b) Consider the elements of l p for 1 p ≤ ∞ ( e n ) j = ² 1 j = n, 0 j 6 = n. Prove that the unit ball in l p is closed and bounded, and use these elements to show that the unit ball is not compact. 2. Find the radius of convergence of the following power series: (a) X n 3 z n , (b) X 2 n n ! z n , 1
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2 (c) X 2 n n 2 z n , (d) X n 3 3 n z n . 3. Suppose that the coefcients oF the power series a n z n are integers, in±nitely many oF which are distinct From zero. Prove that the radius oF convergence is at most 1. 4. Consider a set oF real numbers { a i } p i =0 R , such that a 0 + a 1 + a 2 + ··· + a p = 0 . Prove that
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Unformatted text preview: lim n →∞ ± a √ n + a 1 √ n + 1 + a 2 √ n + 2 + ··· + a p √ n + p ² = 0 . 5. Discuss the convergence behavior oF the Following series stating in each case whether the series is absolutely convergent, conditionally conver-gent, or divergent: (a) ∞ X n =1 1 n ! . (b) ∞ X n =3 n ( n + 1)( n + 2) ( n + 3) 3 . (c) ∞ X n =1 ³ 1 + 1 n 2 ´ n 2 . (d) ∞ X n =1 (-1) n √ n . 3 6. Give an example, if possible, of each of the following. If no example is possible, brieFy give reasons why: (a) A sequence a n → 0, such that ∑ ∞ n =1 a n n diverges. (b) A sequence a n → 0, such that ∑ ∞ n =1 a n n 2 diverges. (c) A series ∑ a n which converges such that ∑ a 2 n diverges....
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hw6 - lim n →∞ ± a √ n a 1 √ n 1 a 2 √ n 2 ··...

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