r N ! N X Q ( N ) ( N k )! N k = + O (1) 2 1 k N AofA - PowerPoint PPT Presentation

“ An equation means nothing to me unless it expresses a thought of God.” − Srinivasa Ramanujan r N ! πN X Q ( N ) ≡ ( N − k )! N k = + O (1) 2 1 ≤ k ≤ N

AofA Asymptotics Q&A 1 � 1 e H 2 N − H N � O Q. Give an asymptotic approximation of to within N 2 A. 3

AofA Asymptotics Q&A 1 (improved version) � 1 1 Q. Match each function with an asymptotic expansion. � 1 + 2 N + O N 2 � 1 1 + 1 � N + O N 2 H N N + O (1) � � exp − 1 H 2 N − H N � 1 1 − 1 � N + O N 2 � � exp H N � 1 1 � 1 − 2 N + O � 1 N 2 � exp N � 1 ln N + γ + 1 � 2 N + O 1 + 1 N 2 � − 1 � � 1 N � N + γ + O N 4

AofA Asymptotics Q&A 2 Q. Match each of the topics described in the book with a mathematician’s name. Approximate a sum with an integral Laplace Expand a differentiable function Ramanujan Approximate factorials Euler Birthday function Taylor Approximate a function by swapping tails Stirling No, this is not high school, but… You do not want to appear to be ignorant ! 5

AofA Asymptotics Q&A 3 1.10102 Q. Match each expression with an approximation to its value. 1.10462 1.01 10 1.22019 1.50034 1.05 10 1.62889 1.01 20 1.64463 1.01 50 2.02300 2.70481 1.01 100 2.71828 6

✓ t ◆ (1 + x ) t = X x k k 0 ≤ k ≤ t = 1 + tx + t ( t − 1) x 2 + O ( x 3 ) 2 ✗ N + α 2 N 2 (1 + 1 N ) αN = 1 + α N � 1 2 N 2 + . . . (1 + 1 N ) t = 1 + t N + t ( t − 1) � + O 2 N 2 N 3 (1 + 1 N ) αN = exp � � αN ln(1 + 1 /N ) 1 . 01 10 = 1 + 10 90 100 + 20000 + . . . αN (1 /N + O (1 /N 2 )) � � = exp � 1 ≈ 1 . 1045 = e α + O � N 1 . 01 50 ≈ √ e

AofA Asymptotics Q&A 3 1.10102 Q. Match each expression with an approximation to its value. 1.10462 1.01 10 1.22019 1.50034 1.05 10 1.62889 1.01 20 1.64463 1.01 50 2.02300 2.70481 1.01 100 2.71828 8