exercises_14 - Analysis 1: Exercises 14 1. Let (a(n) be a...

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Analysis 1: Exercises 14 1. Let ( a ( n )) be a decreasing sequence of strictly positive numbers with n -th partial sum s ( n ). (a) By grouping terms in two different ways, show that 1 2 { a (1) + 2 a (2) + ··· + 2 n a (2 n ) } 6 s (2 n ) 6 ± a (1) + 2 a (2) + ··· + 2 n - 1 a (2 n - 1 ) ² + a (2 n ) . (b) Use (a) to show that X n =1 a ( n ) converges if and only if X n =1 2 n a (2 n ) converges. This result is called the Cauchy condensation test . 2. Let s be the sum of the alternating series n =1 ( - 1) n +1 a n with n -th partial sum s n . Show that | s - s n | 6 a n +1 . 3. Consider the series 1 - 1 2 - 1 3 + 1 4 + 1 5 - 1 6 - 1 7 + ··· where the signs come in pairs. Does it converge? 4. Investigate the convergence of the following series. (a) X n =1 ( - 1) n n p , where p is some real number. (b) X n =1 ( - 1) n n n + 10 . (c) X n =1 x n n , where x is some real number. 5.
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exercises_14 - Analysis 1: Exercises 14 1. Let (a(n) be a...

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