tut02 - (b Set a = x = 2 and use a to compute x 3 Compare...

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The University of Sydney School of Mathematics and Statistics Tutorial 2 (Week 3) MATH2962: Real and Complex Analysis (Advanced) Semester 1, 2009 Web Page: http://www.maths.usyd.edu.au/u/UG/IM/MATH2962/ Lecturer: Daniel Daners Questions to complete during the tutorial 1. Show that the following sequences are monotone. Also check whether they are bounded and therefore convergent. If possible determine the limit. (a) x n = n e + e - n ; (b) s n = n s k =1 1 k ; (c) u 0 > 0, u n +1 = u n r 1 + u 2 n , n N ; (d) v 0 = 0, v n +1 = 3 v n + 4, n N . 2. Determine the limit of the following sequences ( x n ) as n → ∞ if it exists. (a) x n = 2 n 2 + 2 n n 100 + 3 · 2 n ; (b) x n = ( n r n ( n 1) ) ; (c) x n = 1 + 2 + ··· + n n + 2 n 2 ; (d) x n = ( a n + b n ) 1 /n if 0 a b ; 3. (a) Let a > 0. DeFne a sequence recursively by choosing x 0 > 0 arbitrary and setting x n +1 := 1 2 p x n + a x n P for all n N . Prove that ( x n ) n 1 is decreasing and that x n a as n → ∞ . (This is a practical numerical method to compute square roots as ( x n ) converges very fast.)
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Unformatted text preview: (b) Set a = x = 2 and use ( a ) to compute x 3 . Compare the result to √ 2 from your calculator. Extra questions for further practice 4. Let a ∈ C with a n = 1. Use induction by n to prove that s n := n s k =0 a k = 1 − a n +1 1 − a for all n ∈ N . Conclude that ( s n ) converges if and only if | a | < 1. In case of convergence show that s n → (1 − a )-1 as n → ∞ . Challenge questions (optional) * 5. Consider the sequence given by s n := ∑ n k =0 1 k ! . In lectures it is shows that s n → e . (a) Prove that s n < e < s n + 1 n ! n for all n ≥ 1. Determine e to four decimal places. (b) Use ( a ) to show that e is irrational. (Give a proof by contradiction.) Copyright c c 2009 The University of Sydney...
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