MIT6_042JS10_lec25

MIT6_042JS10_lec25 - Closed form for n Mathematics for...

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1 lec 9W.1 Albert R Meyer, April 7, 2010 Asymptotic Notation Mathematics for Computer Science MIT 6.042J/18.062J lec 9W.2 Albert R Meyer, April 7, 2010 Closed form for n! Turn product into a sum taking logs: ln(n!) = ln( 1·2·3···(n – 1)·n ) = ln 1 + ln 2 + · · · + ln(n – 1) + ln(n) lec 9W.3 Albert R Meyer, April 7, 2010 Integral Method to bound ln 2 ln 3 ln 4 ln 5 ln n -1 ln n ln 2 ln 3 ln 4 ln 5 ln n 23 14 5 n –2 n –1 n ln(x+1) ln(x) Closed form for n! lec 9W.5 Albert R Meyer, April 7, 2010 nln n e ! " # $ % & +1 ' ln(i i=1 n ( ) ' (n+1)ln n+1 e ! " # $ % & +0.6 reminder: Closed form for n! lec 9W.6 Albert R Meyer, April 7, 2010 Closed form for n! exponentiating: lec 9W.7 Albert R Meyer, April 7, 2010 n! ~ 2±n n e ' ! ( # $ " n A precise approximation: Stirling’s Formula
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2 lec 9W.13 Albert R Meyer, April 7, 2010 Asymptotically smaller : Def: f(n) = o( g(n) ) iff lim n (' f(n) g(n) = 0 Little Oh: o(·) lec 9W.14 Albert R Meyer, April 7, 2010 Little Oh: o(·) because n 2 = o( n 3 ) lim n ( ± n 2 n 3 = lim n 1 n = 0 lec 9W.15 Albert R Meyer, April 7, 2010 Asymptotic Order of Growth: f(n) = O( g(n) ) limsup n f(n) g(n) !
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This note was uploaded on 05/27/2011 for the course CS 6.042J taught by Professor Prof.albertr.meyer during the Spring '11 term at MIT.

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MIT6_042JS10_lec25 - Closed form for n Mathematics for...

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