Chapter 3 Random Variables Appendix (Exponential Function)

Chapter 3 Random Variables Appendix (Exponential Function)...

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THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT1306 INTRODUCTORY STATISTICS (SEMESTER 1 2008/2009) Exponential Function Now, we show that lim n !1 1 + x n ± n = e x where e is an irrational number. By binomial theorem, we have 1 + x n ± n = n X k =0 n ! ( n k )! k
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Unformatted text preview: ! & x n ± k = n X k =0 n ! n k ( n & k )! x k k ! = n X k =0 n P k n k x k k ! : Taking the limits gives lim n !1 & 1 + x n ± n = 1 X k =0 x k k ! = 1 + x + x 2 2! + x 3 3! + ±±± = e x since n P k n k ! 1 as n ! 1 . 1...
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