exercises_15 - Analysis 1: Exercises 15 1. Suppose that 1...

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Analysis 1: Exercises 15 1. Suppose that 1 a n and 1 b n both converge. Let α, β R . Prove that X 1 ( αa n + β b n ) = α X 1 a n + β X 1 b n . This property of series is called linearity . 2. (a) Show that if x > 0 and if N 3 n > 2 x , then ± ± ± ± e x - ² 1 + x 1! + ··· + x n n ! ³± ± ± ± < 2 x n +1 ( n + 1)! . (b) Use the formula in (a) to show that 2 7 12 < e < 2 3 4 . 3. Show that if 0 6 x 6 a and n N , then 1 + x 1! + ··· + x n n ! 6 e x 6 1 + x 1! + ··· + x n - 1 ( n - 1)! + e a x n n ! . 4. Show that log( 1 + x 1 - x ) = 2 ² x + x 3 3 + x 5 5 + x 7 7 + ··· ³ for | x | < 1. The series is called Gregory’s series . 5. Compute the following limits (a) lim x 0 a x - 1 x where a > 0; (b) lim x 0 log(1 + ax ) x where a > 0; (c) lim x 0 1 - cos x x 2 ; (d) lim x →∞ ´ cos a x µ x 2 , where a > 0. 6. (a) Let f : [0 , ) R be given by f ( x ) = x log x for x > 0 and f (0) = 0. Is f continuous at 0? (b) Let f : [0 , ) R be given by f ( x ) = x 2 log x for x > 0 and f (0) = 0. Is f continuous at 0? Is it right-differentiable at 0? 7. Prove that for x 0 the inequality
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exercises_15 - Analysis 1: Exercises 15 1. Suppose that 1...

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