p11 - solute convergence or divergence of the series a ∞...

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Math 5615H: Introduction to Analysis I. Fall 2011 Homework #11 (due on Wednesday, November 23). 50 points are divided between 4 problems. #1. (12 points). Find S := X n =1 n 2 n . Hint . One can try to rewrite 2 S in terms of S , or use Theorem 3.41 with a n = 1 , b n = 2 - n . #2. (10 points). Give an example of a convergent series a n , such that X n =0 c n diverges, where c n := n X k =0 a k a n - k . #3. (14 points). Investigate the behavior (absolute convergence, nonab-
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Unformatted text preview: solute convergence, or divergence) of the series ( a ) ∞ X n =1 (-1) n ln n n ; ( b ) ∞ X n =1 ln n n 2 . #4. (14 points). Using Euler’s formula e it = cos t + i sin t, find C ( x ) := ∞ X n =0 x n cos nx and S ( x ) := ∞ X n =0 x n sin nx for | x | < 1....
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