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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 Three '09
 daners
 Statistics, Convergence, Mathematical analysis, Xn, University of Sydney School of Mathematics

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