Foundations of Computer Science Lecture 9 Sums And Asymptotics - PowerPoint PPT Presentation

Computing Sums: Tool 1: Constant Rule 10 � S 1 = i =1 3 = 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 3 × 10 Creator: Malik Magdon-Ismail Sums And Asymptotics: 6 / 16 Addition Rule →

Computing Sums: Tool 1: Constant Rule 10 � S 1 = i =1 3 = 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 3 × 10 10 S 2 = i =1 j � Creator: Malik Magdon-Ismail Sums And Asymptotics: 6 / 16 Addition Rule →

Computing Sums: Tool 1: Constant Rule 10 � S 1 = i =1 3 = 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 3 × 10 10 S 2 = i =1 j = j + j + j + j + j + j + j + j + j + j � Creator: Malik Magdon-Ismail Sums And Asymptotics: 6 / 16 Addition Rule →

Computing Sums: Tool 1: Constant Rule 10 � S 1 = i =1 3 = 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 3 × 10 10 S 2 = i =1 j = j + j + j + j + j + j + j + j + j + j � j × 10 Creator: Malik Magdon-Ismail Sums And Asymptotics: 6 / 16 Addition Rule →

Computing Sums: Tool 1: Constant Rule 10 � S 1 = i =1 3 = 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 3 × 10 10 S 2 = i =1 j = j + j + j + j + j + j + j + j + j + j � j × 10 10 � S 3 = i =1 i Creator: Malik Magdon-Ismail Sums And Asymptotics: 6 / 16 Addition Rule →

Computing Sums: Tool 1: Constant Rule 10 � S 1 = i =1 3 = 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 3 × 10 10 S 2 = i =1 j = j + j + j + j + j + j + j + j + j + j � j × 10 10 � S 3 = i =1 i = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 Creator: Malik Magdon-Ismail Sums And Asymptotics: 6 / 16 Addition Rule →

Computing Sums: Tool 1: Constant Rule 10 � S 1 = i =1 3 = 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 3 × 10 10 S 2 = i =1 j = j + j + j + j + j + j + j + j + j + j � j × 10 10 1 � S 3 = i =1 i = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 2 × 10 × (10 + 1) Creator: Malik Magdon-Ismail Sums And Asymptotics: 6 / 16 Addition Rule →

Computing Sums: Tool 1: Constant Rule 10 � S 1 = i =1 3 = 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 3 × 10 10 S 2 = i =1 j = j + j + j + j + j + j + j + j + j + j � j × 10 10 1 � S 3 = i =1 i = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 2 × 10 × (10 + 1) The index of summation is i in these examples. Creator: Malik Magdon-Ismail Sums And Asymptotics: 6 / 16 Addition Rule →

Computing Sums: Tool 1: Constant Rule 10 � S 1 = i =1 3 = 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 3 × 10 10 S 2 = i =1 j = j + j + j + j + j + j + j + j + j + j � j × 10 10 1 � S 3 = i =1 i = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 2 × 10 × (10 + 1) The index of summation is i in these examples. Constants (independent of summation index) can be taken outside the sum. 10 10 10 10 S 1 = i =1 3 = 3 � i =1 1 = 3 × 10 � S 2 = i =1 j = j � i =1 1 = j × 10 . � Creator: Malik Magdon-Ismail Sums And Asymptotics: 6 / 16 Addition Rule →

Computing Sums: Tool 1: Constant Rule 10 � S 1 = i =1 3 = 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 3 × 10 10 S 2 = i =1 j = j + j + j + j + j + j + j + j + j + j � j × 10 10 1 � S 3 = i =1 i = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 2 × 10 × (10 + 1) The index of summation is i in these examples. Constants (independent of summation index) can be taken outside the sum. 10 10 10 10 S 1 = i =1 3 = 3 � i =1 1 = 3 × 10 � S 2 = i =1 j = j � i =1 1 = j × 10 . � n Pop Quiz 9.2 Compute T 4 ( n ) = 5 + i =1 10 . � Creator: Malik Magdon-Ismail Sums And Asymptotics: 6 / 16 Addition Rule →

Computing Sums: Tool 2: Addition Rule 5 i =1 ( i + i 2 ) S = � Creator: Malik Magdon-Ismail Sums And Asymptotics: 7 / 16 Common Sums →

Computing Sums: Tool 2: Addition Rule 5 i =1 ( i + i 2 ) S = � = (1 + 1 2 ) + (2 + 2 2 ) + (3 + 3 2 ) + (4 + 4 2 ) + (5 + 5 2 ) Creator: Malik Magdon-Ismail Sums And Asymptotics: 7 / 16 Common Sums →

Computing Sums: Tool 2: Addition Rule 5 i =1 ( i + i 2 ) S = � = (1 + 1 2 ) + (2 + 2 2 ) + (3 + 3 2 ) + (4 + 4 2 ) + (5 + 5 2 ) = (1 + 2 + 3 + 4 + 5) + (1 2 + 2 2 + 3 2 + 4 2 + 5 2 ) (rearrange terms) Creator: Malik Magdon-Ismail Sums And Asymptotics: 7 / 16 Common Sums →

Computing Sums: Tool 2: Addition Rule 5 i =1 ( i + i 2 ) S = � = (1 + 1 2 ) + (2 + 2 2 ) + (3 + 3 2 ) + (4 + 4 2 ) + (5 + 5 2 ) = (1 + 2 + 3 + 4 + 5) + (1 2 + 2 2 + 3 2 + 4 2 + 5 2 ) (rearrange terms) 5 5 i =1 i 2 . � � = i =1 i + Creator: Malik Magdon-Ismail Sums And Asymptotics: 7 / 16 Common Sums →

Computing Sums: Tool 2: Addition Rule 5 i =1 ( i + i 2 ) S = � = (1 + 1 2 ) + (2 + 2 2 ) + (3 + 3 2 ) + (4 + 4 2 ) + (5 + 5 2 ) = (1 + 2 + 3 + 4 + 5) + (1 2 + 2 2 + 3 2 + 4 2 + 5 2 ) (rearrange terms) 5 5 i =1 i 2 . � � = i =1 i + The sum of terms added together is the addition of the individual sums. � i ( a ( i ) + b ( i ) + c ( i ) + · · · ) = � i a ( i ) + � i b ( i ) + � i c ( i ) + · · · Creator: Malik Magdon-Ismail Sums And Asymptotics: 7 / 16 Common Sums →

Computing Sums: Tool 3: Common Sums n n i =0 2 i = 2 n +1 − 1 n i = k 1 = n + 1 − k � � 1 � i =1 i = 2 n ( n + 1) n 2 i = 2 − 1 1 n i =1 i 2 = n i =1 f ( x ) = nf ( x ) � 1 � 6 n ( n + 1)(2 n + 1) � 2 n i =0 i =0 r i = 1 − r n +1 n i =1 i 3 = n n 4 n 2 ( n + 1) 2 1 � i =1 log i = log n ! � � ( r � =1) 1 − r n i =1 (1 + 2 i + 2 i +2 ) � Example: n i =1 (1 + 2 i + 2 i +2 ) = � Creator: Malik Magdon-Ismail Sums And Asymptotics: 8 / 16 Nested Sums →

Computing Sums: Tool 3: Common Sums n n i =0 2 i = 2 n +1 − 1 n i = k 1 = n + 1 − k � � 1 � i =1 i = 2 n ( n + 1) n 2 i = 2 − 1 1 n i =1 i 2 = n i =1 f ( x ) = nf ( x ) � 1 � 6 n ( n + 1)(2 n + 1) � 2 n i =0 i =0 r i = 1 − r n +1 n i =1 i 3 = n n 1 4 n 2 ( n + 1) 2 � i =1 log i = log n ! � � ( r � =1) 1 − r n i =1 (1 + 2 i + 2 i +2 ) � Example: n n n n i =1 (1 + 2 i + 2 i +2 ) = i =1 2 i +2 � � � � i =1 1 + i =1 2 i + (addition rule) Creator: Malik Magdon-Ismail Sums And Asymptotics: 8 / 16 Nested Sums →

Computing Sums: Tool 3: Common Sums n n i =0 2 i = 2 n +1 − 1 n i = k 1 = n + 1 − k � � 1 � i =1 i = 2 n ( n + 1) n 2 i = 2 − 1 1 n i =1 i 2 = n i =1 f ( x ) = nf ( x ) � 1 � 6 n ( n + 1)(2 n + 1) � 2 n i =0 i =0 r i = 1 − r n +1 n i =1 i 3 = n n 1 4 n 2 ( n + 1) 2 � i =1 log i = log n ! � � ( r � =1) 1 − r n i =1 (1 + 2 i + 2 i +2 ) � Example: n n n n i =1 (1 + 2 i + 2 i +2 ) = i =1 2 i +2 � � � � i =1 1 + i =1 2 i + (addition rule) n n n i =1 2 i � � � = i =1 1 + 2 i =1 i + 4 (constant rule) Creator: Malik Magdon-Ismail Sums And Asymptotics: 8 / 16 Nested Sums →

Computing Sums: Tool 3: Common Sums n n i =0 2 i = 2 n +1 − 1 n i = k 1 = n + 1 − k � � 1 � i =1 i = 2 n ( n + 1) n 2 i = 2 − 1 1 n i =1 i 2 = n i =1 f ( x ) = nf ( x ) � 1 � 6 n ( n + 1)(2 n + 1) � 2 n i =0 i =0 r i = 1 − r n +1 n i =1 i 3 = n n 4 n 2 ( n + 1) 2 1 � i =1 log i = log n ! � � ( r � =1) 1 − r n i =1 (1 + 2 i + 2 i +2 ) � Example: n n n n i =1 (1 + 2 i + 2 i +2 ) = i =1 2 i +2 � � � � i =1 1 + i =1 2 i + (addition rule) n n n i =1 2 i � � � = i =1 1 + 2 i =1 i + 4 (constant rule) 2 n ( n + 1) + 4 · (2 n +1 − 1 − 1 ) = n + 2 × 1 (common sums) Creator: Malik Magdon-Ismail Sums And Asymptotics: 8 / 16 Nested Sums →

Computing Sums: Tool 3: Common Sums n n i =0 2 i = 2 n +1 − 1 n i = k 1 = n + 1 − k � � 1 � i =1 i = 2 n ( n + 1) n 2 i = 2 − 1 1 n i =1 i 2 = n i =1 f ( x ) = nf ( x ) � 1 � 6 n ( n + 1)(2 n + 1) � 2 n i =0 i =0 r i = 1 − r n +1 n i =1 i 3 = n n 4 n 2 ( n + 1) 2 1 � i =1 log i = log n ! � � ( r � =1) 1 − r n i =1 (1 + 2 i + 2 i +2 ) � Example: n n n n i =1 (1 + 2 i + 2 i +2 ) = i =1 2 i +2 � � � � i =1 1 + i =1 2 i + (addition rule) n n n i =1 2 i � � � = i =1 1 + 2 i =1 i + 4 (constant rule) 2 n ( n + 1) + 4 · (2 n +1 − 1 − 1 ) = n + 2 × 1 (common sums) = n + n ( n + 1) + 2 n +3 − 8 (common sums) Creator: Malik Magdon-Ismail Sums And Asymptotics: 8 / 16 Nested Sums →

Computing Sums: Tool 3: Nested Sum Rule 3 3 3 i � � � � S 1 = j =1 1; S 2 = j =1 1 . i =1 i =1 Creator: Malik Magdon-Ismail Sums And Asymptotics: 9 / 16 Computing T 2 ( n ) →

Computing Sums: Tool 3: Nested Sum Rule 3 3 3 i � � � � S 1 = j =1 1; S 2 = j =1 1 . i =1 i =1 To compute a nested sum, start with the innermost sum and proceed outward. Creator: Malik Magdon-Ismail Sums And Asymptotics: 9 / 16 Computing T 2 ( n ) →

Computing Sums: Tool 3: Nested Sum Rule 3 3 3 i � � � � S 1 = j =1 1; S 2 = j =1 1 . i =1 i =1 To compute a nested sum, start with the innermost sum and proceed outward. 3 3 3 S 1 j =1 1 � j =1 1 � j =1 1 � ( i =1) ( i =2) ( i =3) Creator: Malik Magdon-Ismail Sums And Asymptotics: 9 / 16 Computing T 2 ( n ) →

Computing Sums: Tool 3: Nested Sum Rule 3 3 3 i � � � � S 1 = j =1 1; S 2 = j =1 1 . i =1 i =1 To compute a nested sum, start with the innermost sum and proceed outward. 3 3 3 S 1 = j =1 1 + � j =1 1 + � j =1 1 � ( i =1) ( i =2) ( i =3) Creator: Malik Magdon-Ismail Sums And Asymptotics: 9 / 16 Computing T 2 ( n ) →

Computing Sums: Tool 3: Nested Sum Rule 3 3 3 i � � � � S 1 = j =1 1; S 2 = j =1 1 . i =1 i =1 To compute a nested sum, start with the innermost sum and proceed outward. 3 3 3 S 1 = j =1 1 + � j =1 1 + � j =1 1 � = 3 + 3 + 3 = 9 . ( i =1) ( i =2) ( i =3) Creator: Malik Magdon-Ismail Sums And Asymptotics: 9 / 16 Computing T 2 ( n ) →

Computing Sums: Tool 3: Nested Sum Rule 3 3 3 i � � � � S 1 = j =1 1; S 2 = j =1 1 . i =1 i =1 To compute a nested sum, start with the innermost sum and proceed outward. 3 3 3 S 1 = j =1 1 + � j =1 1 + � j =1 1 � = 3 + 3 + 3 = 9 . ( i =1) ( i =2) ( i =3) 1 2 3 S 2 j =1 1 � j =1 1 � j =1 1 � ( i =1) ( i =2) ( i =3) Creator: Malik Magdon-Ismail Sums And Asymptotics: 9 / 16 Computing T 2 ( n ) →

Computing Sums: Tool 3: Nested Sum Rule 3 3 3 i � � � � S 1 = j =1 1; S 2 = j =1 1 . i =1 i =1 To compute a nested sum, start with the innermost sum and proceed outward. 3 3 3 S 1 = j =1 1 + � j =1 1 + � j =1 1 � = 3 + 3 + 3 = 9 . ( i =1) ( i =2) ( i =3) 1 2 3 S 2 = j =1 1 + � j =1 1 + � j =1 1 � ( i =1) ( i =2) ( i =3) Creator: Malik Magdon-Ismail Sums And Asymptotics: 9 / 16 Computing T 2 ( n ) →

Computing Sums: Tool 3: Nested Sum Rule 3 3 3 i � � � � S 1 = j =1 1; S 2 = j =1 1 . i =1 i =1 To compute a nested sum, start with the innermost sum and proceed outward. 3 3 3 S 1 = j =1 1 + � j =1 1 + � j =1 1 � = 3 + 3 + 3 = 9 . ( i =1) ( i =2) ( i =3) 1 2 3 S 2 = j =1 1 + � j =1 1 + � j =1 1 � = 1 + 2 + 3 = 6 . ( i =1) ( i =2) ( i =3) Creator: Malik Magdon-Ismail Sums And Asymptotics: 9 / 16 Computing T 2 ( n ) →

Computing Sums: Tool 3: Nested Sum Rule 3 3 3 i � � � � S 1 = j =1 1; S 2 = j =1 1 . i =1 i =1 To compute a nested sum, start with the innermost sum and proceed outward. 3 3 3 S 1 = j =1 1 + � j =1 1 + � j =1 1 � = 3 + 3 + 3 = 9 . ( i =1) ( i =2) ( i =3) 1 2 3 S 2 = j =1 1 + � j =1 1 + � j =1 1 � = 1 + 2 + 3 = 6 . ( i =1) ( i =2) ( i =3) More generally: n i � � S ( n ) = j =1 1 i =1 Creator: Malik Magdon-Ismail Sums And Asymptotics: 9 / 16 Computing T 2 ( n ) →

Computing Sums: Tool 3: Nested Sum Rule 3 3 3 i � � � � S 1 = j =1 1; S 2 = j =1 1 . i =1 i =1 To compute a nested sum, start with the innermost sum and proceed outward. 3 3 3 S 1 = j =1 1 + � j =1 1 + � j =1 1 � = 3 + 3 + 3 = 9 . ( i =1) ( i =2) ( i =3) 1 2 3 S 2 = j =1 1 + � j =1 1 + � j =1 1 � = 1 + 2 + 3 = 6 . ( i =1) ( i =2) ( i =3) More generally: n i n i � � � � S ( n ) = j =1 1 = j =1 1 i =1 i =1 � �� � f ( i )= i Creator: Malik Magdon-Ismail Sums And Asymptotics: 9 / 16 Computing T 2 ( n ) →

Computing Sums: Tool 3: Nested Sum Rule 3 3 3 i � � � � S 1 = j =1 1; S 2 = j =1 1 . i =1 i =1 To compute a nested sum, start with the innermost sum and proceed outward. 3 3 3 S 1 = j =1 1 + � j =1 1 + � j =1 1 � = 3 + 3 + 3 = 9 . ( i =1) ( i =2) ( i =3) 1 2 3 S 2 = j =1 1 + � j =1 1 + � j =1 1 � = 1 + 2 + 3 = 6 . ( i =1) ( i =2) ( i =3) More generally: n i n i n � � � � � S ( n ) = j =1 1 = j =1 1 = i =1 i i =1 i =1 � �� � f ( i )= i Creator: Malik Magdon-Ismail Sums And Asymptotics: 9 / 16 Computing T 2 ( n ) →

Computing Sums: Tool 3: Nested Sum Rule 3 3 3 i � � � � S 1 = j =1 1; S 2 = j =1 1 . i =1 i =1 To compute a nested sum, start with the innermost sum and proceed outward. 3 3 3 S 1 = j =1 1 + � j =1 1 + � j =1 1 � = 3 + 3 + 3 = 9 . ( i =1) ( i =2) ( i =3) 1 2 3 S 2 = j =1 1 + � j =1 1 + � j =1 1 � = 1 + 2 + 3 = 6 . ( i =1) ( i =2) ( i =3) More generally: n i n i n i =1 i = 1 � � � � � S ( n ) = j =1 1 = j =1 1 = 2 n ( n + 1) . i =1 i =1 � �� � f ( i )= i Creator: Malik Magdon-Ismail Sums And Asymptotics: 9 / 16 Computing T 2 ( n ) →

n n   Computing a Formula for T 2 ( n ) = 2 +  3 + j = i 6 � �  i =1   n n  = T 2 ( n ) = 2 + �  3 + j = i 6 � i =1 Creator: Malik Magdon-Ismail Sums And Asymptotics: 10 / 16 Practice →

n n   Computing a Formula for T 2 ( n ) = 2 +  3 + j = i 6 � �  i =1   n n n n n  = 2 + T 2 ( n ) = 2 + �  3 + j = i 6 � i =1 3 + � � j = i 6 � (sum rule) i =1 i =1 Creator: Malik Magdon-Ismail Sums And Asymptotics: 10 / 16 Practice →

n n   Computing a Formula for T 2 ( n ) = 2 +  3 + j = i 6 � �  i =1   n n n n n  = 2 + T 2 ( n ) = 2 + �  3 + j = i 6 � i =1 3 + � � j = i 6 � (sum rule) i =1 i =1 n n n � � � = 2 + 3 i =1 1 + j = i 6 (constant rule) i =1 Creator: Malik Magdon-Ismail Sums And Asymptotics: 10 / 16 Practice →

n n   Computing a Formula for T 2 ( n ) = 2 +  3 + j = i 6 � �  i =1   n n n n n  = 2 + T 2 ( n ) = 2 + �  3 + j = i 6 � i =1 3 + � � j = i 6 � (sum rule) i =1 i =1 n n n � � � = 2 + 3 i =1 1 + j = i 6 (constant rule) i =1 n n = 2 + 3 n + � j = i 6 � (common sum) i =1 Creator: Malik Magdon-Ismail Sums And Asymptotics: 10 / 16 Practice →

n n   Computing a Formula for T 2 ( n ) = 2 +  3 + j = i 6 � �  i =1   n n n n n  = 2 + T 2 ( n ) = 2 + �  3 + j = i 6 � i =1 3 + � � j = i 6 � (sum rule) i =1 i =1 n n n � � � = 2 + 3 i =1 1 + j = i 6 (constant rule) i =1 n n = 2 + 3 n + � j = i 6 � (common sum) i =1 n n = 2 + 3 n + � j = i 6 � (innermost sum) i =1 Creator: Malik Magdon-Ismail Sums And Asymptotics: 10 / 16 Practice →

n n   Computing a Formula for T 2 ( n ) = 2 +  3 + j = i 6 � �  i =1   n n n n n  = 2 + T 2 ( n ) = 2 + �  3 + j = i 6 � i =1 3 + � � j = i 6 � (sum rule) i =1 i =1 n n n � � � = 2 + 3 i =1 1 + j = i 6 (constant rule) i =1 n n = 2 + 3 n + � j = i 6 � (common sum) i =1 n n = 2 + 3 n + � j = i 6 � (innermost sum) i =1 n n � � = 2 + 3 n + 6 j = i 1 (constant rule) i =1 Creator: Malik Magdon-Ismail Sums And Asymptotics: 10 / 16 Practice →

n n   Computing a Formula for T 2 ( n ) = 2 +  3 + j = i 6 � �  i =1   n n n n n  = 2 + T 2 ( n ) = 2 + �  3 + j = i 6 � i =1 3 + � � j = i 6 � (sum rule) i =1 i =1 n n n � � � = 2 + 3 i =1 1 + j = i 6 (constant rule) i =1 n n = 2 + 3 n + � j = i 6 � (common sum) i =1 n n = 2 + 3 n + � j = i 6 � (innermost sum) i =1 n n � � = 2 + 3 n + 6 j = i 1 (constant rule) i =1 n = 2 + 3 n + 6 i =1 ( n + 1 − i ) � (common sum) Creator: Malik Magdon-Ismail Sums And Asymptotics: 10 / 16 Practice →

n n   Computing a Formula for T 2 ( n ) = 2 +  3 + j = i 6 � �  i =1   n n n n n  = 2 + T 2 ( n ) = 2 + �  3 + j = i 6 � i =1 3 + � � j = i 6 � (sum rule) i =1 i =1 n n n � � � = 2 + 3 i =1 1 + j = i 6 (constant rule) i =1 n n = 2 + 3 n + � j = i 6 � (common sum) i =1 n n = 2 + 3 n + � j = i 6 � (innermost sum) i =1 n n � � = 2 + 3 n + 6 j = i 1 (constant rule) i =1 n = 2 + 3 n + 6 i =1 ( n + 1 − i ) � (common sum) = 2 + 3 n + 6( n + ( n − 1) + · · · + 1) Creator: Malik Magdon-Ismail Sums And Asymptotics: 10 / 16 Practice →

n n   Computing a Formula for T 2 ( n ) = 2 +  3 + j = i 6 � �  i =1   n n n n n  = 2 + T 2 ( n ) = 2 + �  3 + j = i 6 � i =1 3 + � � j = i 6 � (sum rule) i =1 i =1 n n n � � � = 2 + 3 i =1 1 + j = i 6 (constant rule) i =1 n n = 2 + 3 n + � j = i 6 � (common sum) i =1 n n = 2 + 3 n + � j = i 6 � (innermost sum) i =1 n n � � = 2 + 3 n + 6 j = i 1 (constant rule) i =1 n = 2 + 3 n + 6 i =1 ( n + 1 − i ) � (common sum) = 2 + 3 n + 6( n + ( n − 1) + · · · + 1) = 2 + 3 n + 6 × 1 2 n ( n + 1) (common sum) Creator: Malik Magdon-Ismail Sums And Asymptotics: 10 / 16 Practice →

n n   Computing a Formula for T 2 ( n ) = 2 +  3 + j = i 6 � �  i =1   n n n n n  = 2 + T 2 ( n ) = 2 + �  3 + j = i 6 � i =1 3 + � � j = i 6 � (sum rule) i =1 i =1 n n n � � � = 2 + 3 i =1 1 + j = i 6 (constant rule) i =1 n n = 2 + 3 n + � j = i 6 � (common sum) i =1 n n = 2 + 3 n + � j = i 6 � (innermost sum) i =1 n n � � = 2 + 3 n + 6 j = i 1 (constant rule) i =1 n = 2 + 3 n + 6 i =1 ( n + 1 − i ) � (common sum) = 2 + 3 n + 6( n + ( n − 1) + · · · + 1) = 2 + 3 n + 6 × 1 2 n ( n + 1) (common sum) = 2 + 6 n + 3 n 2 (algebra) Creator: Malik Magdon-Ismail Sums And Asymptotics: 10 / 16 Practice →

n i Practice: Compute a Formula for the Sum j =1 ij � � i =1 n i � j =1 ij = � i =1 Creator: Malik Magdon-Ismail Sums And Asymptotics: 11 / 16 Summary of Max. Substring Sum →

n i Practice: Compute a Formula for the Sum j =1 ij � � i =1 n i n i � j =1 ij = � � j =1 ij � (innermost sum) i =1 i =1 Creator: Malik Magdon-Ismail Sums And Asymptotics: 11 / 16 Summary of Max. Substring Sum →

n i Practice: Compute a Formula for the Sum j =1 ij � � i =1 n i n i � j =1 ij = � � j =1 ij � (innermost sum) i =1 i =1 n i � � = i =1 i j =1 j (constant rule) Creator: Malik Magdon-Ismail Sums And Asymptotics: 11 / 16 Summary of Max. Substring Sum →

n i Practice: Compute a Formula for the Sum j =1 ij � � i =1 n i n i � j =1 ij = � � j =1 ij � (innermost sum) i =1 i =1 n i � � = i =1 i j =1 j (constant rule) n i =1 i × 1 = � 2 i ( i + 1) (common sum) Creator: Malik Magdon-Ismail Sums And Asymptotics: 11 / 16 Summary of Max. Substring Sum →

n i Practice: Compute a Formula for the Sum j =1 ij � � i =1 n i n i � j =1 ij = � � j =1 ij � (innermost sum) i =1 i =1 n i � � = i =1 i j =1 j (constant rule) n i =1 i × 1 = � 2 i ( i + 1) (common sum) n i =1 ( i 3 + i 2 ) 1 = � (algebra, constant rule) 2 Creator: Malik Magdon-Ismail Sums And Asymptotics: 11 / 16 Summary of Max. Substring Sum →

n i Practice: Compute a Formula for the Sum j =1 ij � � i =1 n i n i � j =1 ij = � � j =1 ij � (innermost sum) i =1 i =1 n i � � = i =1 i j =1 j (constant rule) n i =1 i × 1 = � 2 i ( i + 1) (common sum) n i =1 ( i 3 + i 2 ) 1 = � (algebra, constant rule) 2 n n i =1 i 3 + 1 1 i =1 i 2 = � � (sum rule) 2 2 Creator: Malik Magdon-Ismail Sums And Asymptotics: 11 / 16 Summary of Max. Substring Sum →

n i Practice: Compute a Formula for the Sum j =1 ij � � i =1 n i n i � j =1 ij = � � j =1 ij � (innermost sum) i =1 i =1 n i � � = i =1 i j =1 j (constant rule) n i =1 i × 1 = � 2 i ( i + 1) (common sum) n i =1 ( i 3 + i 2 ) 1 = � (algebra, constant rule) 2 n n i =1 i 3 + 1 1 i =1 i 2 = � � (sum rule) 2 2 8 n 2 ( n + 1) 2 + 1 1 = 12 n ( n + 1)(2 n + 1) (common sums) Creator: Malik Magdon-Ismail Sums And Asymptotics: 11 / 16 Summary of Max. Substring Sum →

n i Practice: Compute a Formula for the Sum j =1 ij � � i =1 n i n i � j =1 ij = � � j =1 ij � (innermost sum) i =1 i =1 n i � � = i =1 i j =1 j (constant rule) n i =1 i × 1 = � 2 i ( i + 1) (common sum) n i =1 ( i 3 + i 2 ) 1 = � (algebra, constant rule) 2 n n i =1 i 3 + 1 1 i =1 i 2 = � � (sum rule) 2 2 8 n 2 ( n + 1) 2 + 1 1 = 12 n ( n + 1)(2 n + 1) (common sums) 8 n 2 + 5 12 n 3 + 1 8 n 4 12 n + 3 1 = (algebra) Creator: Malik Magdon-Ismail Sums And Asymptotics: 11 / 16 Summary of Max. Substring Sum →

Summary of Maximum Substring Sum Algorithms 100 Runtimes T 1 ( n ) T 2 ( n ) 2 n 2 + 1 (Running Time)/n 80 T 1 ( n ) = 2 + 31 6 n + 7 3 n 3 60 T 2 ( n ) = 2 + 6 n + 3 n 2 T 3 ( n ) 40 3 n (log 2 n + 1) − 9 ≤ T 3 ( n ) ≤ 12 n (log 2 n + 3) − 9 20 T 4 ( n ) T 4 ( n ) = 5 + 10 n 0 10 20 30 40 50 n (“simple” formulas for T 1 ( n ) , . . . , T 4 ( n )) Creator: Malik Magdon-Ismail Sums And Asymptotics: 12 / 16 Linear Functions Θ( n ) →

Summary of Maximum Substring Sum Algorithms 100 Runtimes T 1 ( n ) T 2 ( n ) 2 n 2 + 1 (Running Time)/n 80 T 1 ( n ) = 2 + 31 6 n + 7 3 n 3 60 T 2 ( n ) = 2 + 6 n + 3 n 2 T 3 ( n ) 40 3 n (log 2 n + 1) − 9 ≤ T 3 ( n ) ≤ 12 n (log 2 n + 3) − 9 20 T 4 ( n ) T 4 ( n ) = 5 + 10 n 0 10 20 30 40 50 n (“simple” formulas for T 1 ( n ) , . . . , T 4 ( n )) So, which algorithm is best? Creator: Malik Magdon-Ismail Sums And Asymptotics: 12 / 16 Linear Functions Θ( n ) →

Summary of Maximum Substring Sum Algorithms 100 Runtimes T 1 ( n ) T 2 ( n ) 2 n 2 + 1 (Running Time)/n 80 T 1 ( n ) = 2 + 31 6 n + 7 3 n 3 60 T 2 ( n ) = 2 + 6 n + 3 n 2 T 3 ( n ) 40 3 n (log 2 n + 1) − 9 ≤ T 3 ( n ) ≤ 12 n (log 2 n + 3) − 9 20 T 4 ( n ) T 4 ( n ) = 5 + 10 n 0 10 20 30 40 50 n (“simple” formulas for T 1 ( n ) , . . . , T 4 ( n )) So, which algorithm is best? Computers solve problems with big inputs. We care about large n . Creator: Malik Magdon-Ismail Sums And Asymptotics: 12 / 16 Linear Functions Θ( n ) →

Summary of Maximum Substring Sum Algorithms 100 Runtimes T 1 ( n ) T 2 ( n ) 2 n 2 + 1 (Running Time)/n 80 T 1 ( n ) = 2 + 31 6 n + 7 3 n 3 60 T 2 ( n ) = 2 + 6 n + 3 n 2 T 3 ( n ) 40 3 n (log 2 n + 1) − 9 ≤ T 3 ( n ) ≤ 12 n (log 2 n + 3) − 9 20 T 4 ( n ) T 4 ( n ) = 5 + 10 n 0 10 20 30 40 50 n (“simple” formulas for T 1 ( n ) , . . . , T 4 ( n )) So, which algorithm is best? Computers solve problems with big inputs. We care about large n . Compare runtimes asymptotically in the input size n . That is n → ∞ . Creator: Malik Magdon-Ismail Sums And Asymptotics: 12 / 16 Linear Functions Θ( n ) →

Summary of Maximum Substring Sum Algorithms 100 Runtimes T 1 ( n ) T 2 ( n ) 2 n 2 + 1 (Running Time)/n 80 T 1 ( n ) = 2 + 31 6 n + 7 3 n 3 60 T 2 ( n ) = 2 + 6 n + 3 n 2 T 3 ( n ) 40 3 n (log 2 n + 1) − 9 ≤ T 3 ( n ) ≤ 12 n (log 2 n + 3) − 9 20 T 4 ( n ) T 4 ( n ) = 5 + 10 n 0 10 20 30 40 50 n (“simple” formulas for T 1 ( n ) , . . . , T 4 ( n )) So, which algorithm is best? Computers solve problems with big inputs. We care about large n . Compare runtimes asymptotically in the input size n . That is n → ∞ . Ignore additive and multiplicative constants (minutia). We care about growth rate . Creator: Malik Magdon-Ismail Sums And Asymptotics: 12 / 16 Linear Functions Θ( n ) →

Summary of Maximum Substring Sum Algorithms 100 Runtimes T 1 ( n ) T 2 ( n ) 2 n 2 + 1 (Running Time)/n 80 T 1 ( n ) = 2 + 31 6 n + 7 3 n 3 60 T 2 ( n ) = 2 + 6 n + 3 n 2 T 3 ( n ) 40 3 n (log 2 n + 1) − 9 ≤ T 3 ( n ) ≤ 12 n (log 2 n + 3) − 9 20 T 4 ( n ) T 4 ( n ) = 5 + 10 n 0 10 20 30 40 50 n (“simple” formulas for T 1 ( n ) , . . . , T 4 ( n )) So, which algorithm is best? Computers solve problems with big inputs. We care about large n . Compare runtimes asymptotically in the input size n . That is n → ∞ . Ignore additive and multiplicative constants (minutia). We care about growth rate . Algorithm 4 is linear in n , T 4 ( n ) → constant. n Creator: Malik Magdon-Ismail Sums And Asymptotics: 12 / 16 Linear Functions Θ( n ) →

Asymptotically Linear Functions: Θ( n ), big-Theta-of- n T ∈ Θ( n ) , if there are positive constants c, C for which c · n ≤ T ( n ) ≤ C · n . Creator: Malik Magdon-Ismail Sums And Asymptotics: 13 / 16 General Asymptotics →

Asymptotically Linear Functions: Θ( n ), big-Theta-of- n T ∈ Θ( n ) , if there are positive constants c, C for which c · n ≤ T ( n ) ≤ C · n .  “ T > n ” ; T ∈ ω ( n ) ,  ∞       T ( n )  − →    constant>0 “ T = n ” ; T ∈ Θ( n ) , n →∞ n        “ T < n ” . 0 T ∈ o ( n ) ,     Creator: Malik Magdon-Ismail Sums And Asymptotics: 13 / 16 General Asymptotics →

Asymptotically Linear Functions: Θ( n ), big-Theta-of- n T ∈ Θ( n ) , if there are positive constants c, C for which c · n ≤ T ( n ) ≤ C · n .  “ T > n ” ; T ∈ ω ( n ) ,  ∞       T ( n )  − →    constant>0 “ T = n ” ; T ∈ Θ( n ) , n →∞ n        “ T < n ” . 0 T ∈ o ( n ) ,     Linear means in Θ( n ) : 2 n + 15 √ n, 10 9 n + 3 , 2 log 2 n +4 . 2 n + 7 , 3 n + log n, Creator: Malik Magdon-Ismail Sums And Asymptotics: 13 / 16 General Asymptotics →

Asymptotically Linear Functions: Θ( n ), big-Theta-of- n T ∈ Θ( n ) , if there are positive constants c, C for which c · n ≤ T ( n ) ≤ C · n .  “ T > n ” ; T ∈ ω ( n ) ,  ∞       T ( n )  − →    constant>0 “ T = n ” ; T ∈ Θ( n ) , n →∞ n        “ T < n ” . 0 T ∈ o ( n ) ,     Linear means in Θ( n ) : 2 n + 15 √ n, 10 9 n + 3 , 2 log 2 n +4 . 2 n + 7 , 3 n + log n, Not linear, not in Θ( n ) : 10 9 √ n + 15 , n 10 − 9 n 2 , n 1 . 0001 , n 0 . 9999 , 2 n . n log n, log n, Creator: Malik Magdon-Ismail Sums And Asymptotics: 13 / 16 General Asymptotics →

Asymptotically Linear Functions: Θ( n ), big-Theta-of- n T ∈ Θ( n ) , if there are positive constants c, C for which c · n ≤ T ( n ) ≤ C · n .  “ T > n ” ; T ∈ ω ( n ) ,  ∞       T ( n )  − →    constant>0 “ T = n ” ; T ∈ Θ( n ) , n →∞ n        “ T < n ” . 0 T ∈ o ( n ) ,     Linear means in Θ( n ) : 2 n + 15 √ n, 10 9 n + 3 , 2 log 2 n +4 . 2 n + 7 , 3 n + log n, Not linear, not in Θ( n ) : 10 9 √ n + 15 , n 10 − 9 n 2 , n 1 . 0001 , n 0 . 9999 , 2 n . n log n, log n, Other runtimes from practice: log linear loglinear quadratic cubic superpolynomial exponential factorial BAD Θ( n 2 ) Θ( n 3 ) Θ( n log n ) Θ(2 n ) Θ( n n ) Θ(log n ) Θ( n ) Θ( n log n ) Θ( n !) Creator: Malik Magdon-Ismail Sums And Asymptotics: 13 / 16 General Asymptotics →

General Asymptotics: Θ( f ), big-Theta-of- f ω ( f )  T ∈ ω ( f ) , “ T < f ” ;  ∞  T ( n ) /f ( n )      T ( n )  Θ( f ) f ( n ) − →    constant>0 T ∈ Θ( f ) , “ T = f ” ; n →∞        0 T ∈ o ( f ) , “ T < f ” .   o ( f )   n − → T ∈ o ( f ) T ∈ Θ( f ) T ∈ ω ( f ) “ T < f ” “ T = f ” “ T > f ” cf ( n ) ≤ T ( n ) ≤ Cf ( n ) Examples and Practice. (See also Exercise 9.6) Creator: Malik Magdon-Ismail Sums And Asymptotics: 14 / 16 The Integration Method →

General Asymptotics: Θ( f ), big-Theta-of- f ω ( f )  T ∈ ω ( f ) , “ T < f ” ;  ∞  T ( n ) /f ( n )      T ( n )  Θ( f ) f ( n ) − →    constant>0 T ∈ Θ( f ) , “ T = f ” ; n →∞        0 T ∈ o ( f ) , “ T < f ” .   o ( f )   n − → T ∈ o ( f ) T ∈ Θ( f ) T ∈ ω ( f ) “ T < f ” “ T ≤ f ” “ T = f ” “ T ≥ f ” “ T > f ” cf ( n ) ≤ T ( n ) ≤ Cf ( n ) Examples and Practice. (See also Exercise 9.6) Creator: Malik Magdon-Ismail Sums And Asymptotics: 14 / 16 The Integration Method →

General Asymptotics: Θ( f ), big-Theta-of- f ω ( f )  T ∈ ω ( f ) , “ T < f ” ;  ∞  T ( n ) /f ( n )      T ( n )  Θ( f ) f ( n ) − →    constant>0 T ∈ Θ( f ) , “ T = f ” ; n →∞        0 T ∈ o ( f ) , “ T < f ” .   o ( f )   n − → T ∈ o ( f ) T ∈ O ( f ) T ∈ Θ( f ) T ∈ Ω( f ) T ∈ ω ( f ) “ T < f ” “ T ≤ f ” “ T = f ” “ T ≥ f ” “ T > f ” cf ( n ) ≤ T ( n ) ≤ Cf ( n ) Examples and Practice. (See also Exercise 9.6) Creator: Malik Magdon-Ismail Sums And Asymptotics: 14 / 16 The Integration Method →

General Asymptotics: Θ( f ), big-Theta-of- f ω ( f )  T ∈ ω ( f ) , “ T < f ” ;  ∞  T ( n ) /f ( n )      T ( n )  Θ( f ) f ( n ) − →    constant>0 T ∈ Θ( f ) , “ T = f ” ; n →∞        0 T ∈ o ( f ) , “ T < f ” .   o ( f )   n − → T ∈ o ( f ) T ∈ O ( f ) T ∈ Θ( f ) T ∈ Ω( f ) T ∈ ω ( f ) “ T < f ” “ T ≤ f ” “ T = f ” “ T ≥ f ” “ T > f ” T ( n ) ≤ Cf ( n ) cf ( n ) ≤ T ( n ) ≤ Cf ( n ) cf ( n ) ≤ T ( n ) Examples and Practice. (See also Exercise 9.6) Creator: Malik Magdon-Ismail Sums And Asymptotics: 14 / 16 The Integration Method →

General Asymptotics: Θ( f ), big-Theta-of- f ω ( f )  T ∈ ω ( f ) , “ T < f ” ;  ∞  T ( n ) /f ( n )      T ( n )  Θ( f ) f ( n ) − →    constant>0 T ∈ Θ( f ) , “ T = f ” ; n →∞        0 T ∈ o ( f ) , “ T < f ” .   o ( f )   n − → T ∈ o ( f ) T ∈ O ( f ) T ∈ Θ( f ) T ∈ Ω( f ) T ∈ ω ( f ) “ T < f ” “ T ≤ f ” “ T = f ” “ T ≥ f ” “ T > f ” T ( n ) ≤ Cf ( n ) cf ( n ) ≤ T ( n ) ≤ Cf ( n ) cf ( n ) ≤ T ( n ) Examples and Practice. (See also Exercise 9.6) For polynomials, growth rate is the highest order. n 2 + n √ n n 2 + log 256 n n 2 + n 1 . 99 log 256 n 2 n 2 Θ( n 2 ) Θ( n 2 ) Θ( n 2 ) Θ( n 2 ) Creator: Malik Magdon-Ismail Sums And Asymptotics: 14 / 16 The Integration Method →

General Asymptotics: Θ( f ), big-Theta-of- f ω ( f )  T ∈ ω ( f ) , “ T < f ” ;  ∞  T ( n ) /f ( n )      T ( n )  Θ( f ) f ( n ) − →    constant>0 T ∈ Θ( f ) , “ T = f ” ; n →∞        0 T ∈ o ( f ) , “ T < f ” .   o ( f )   n − → T ∈ o ( f ) T ∈ O ( f ) T ∈ Θ( f ) T ∈ Ω( f ) T ∈ ω ( f ) “ T < f ” “ T ≤ f ” “ T = f ” “ T ≥ f ” “ T > f ” T ( n ) ≤ Cf ( n ) cf ( n ) ≤ T ( n ) ≤ Cf ( n ) cf ( n ) ≤ T ( n ) Examples and Practice. (See also Exercise 9.6) For polynomials, growth rate is the highest order. For nested sums, growth rate is number of nestings plus order of summand. n 2 + n √ n n 2 + log 256 n n 2 + n 1 . 99 log 256 n n n i n i 2 n 2 � � � � � i =1 i j =1 1 j =1 ij i =1 i =1 Θ( n 2 ) Θ( n 2 ) Θ( n 2 ) Θ( n 2 ) Θ( n 2 ) Θ( n 2 ) Θ( n 4 ) Creator: Malik Magdon-Ismail Sums And Asymptotics: 14 / 16 The Integration Method →

The Integration Method � n 0 dx f ( x ) f ( x ) f ( n ) · · · f (3) f (2) f (1) 0 1 2 3 · · · n n + 1 Theorem. (Integration Bound) For a monotonically increasing function f , Creator: Malik Magdon-Ismail Sums And Asymptotics: 15 / 16 Integration For Asymptotic Behavior →

The Integration Method � n 0 dx f ( x ) f ( x ) f ( n ) · · · f (3) f (2) f (1) 0 1 2 3 · · · n n + 1 Theorem. (Integration Bound) For a monotonically increasing function f , � n n m − 1 dx f ( x ) ≤ i = m f ( i ) � Creator: Malik Magdon-Ismail Sums And Asymptotics: 15 / 16 Integration For Asymptotic Behavior →

The Integration Method � n � n +1 0 dx f ( x ) dx f ( x ) 1 f ( x ) f ( n ) f ( n ) · · · · · · f (3) f (3) f (2) f (2) f (1) f (1) 0 1 2 3 · · · n n + 1 0 1 2 3 n + 1 · · · n Theorem. (Integration Bound) For a monotonically increasing function f , � n n m − 1 dx f ( x ) ≤ i = m f ( i ) � Creator: Malik Magdon-Ismail Sums And Asymptotics: 15 / 16 Integration For Asymptotic Behavior →

The Integration Method � n � n +1 0 dx f ( x ) dx f ( x ) 1 f ( x ) f ( n ) f ( n ) · · · · · · f (3) f (3) f (2) f (2) f (1) f (1) 0 1 2 3 · · · n n + 1 0 1 2 3 n + 1 · · · n Theorem. (Integration Bound) For a monotonically increasing function f , � n � n +1 n m − 1 dx f ( x ) ≤ i = m f ( i ) ≤ � dx f ( x ) . m Creator: Malik Magdon-Ismail Sums And Asymptotics: 15 / 16 Integration For Asymptotic Behavior →

The Integration Method � n � n +1 0 dx f ( x ) dx f ( x ) 1 f ( x ) f ( n ) f ( n ) · · · · · · f (3) f (3) f (2) f (2) f (1) f (1) 0 1 2 3 · · · n n + 1 0 1 2 3 n + 1 · · · n Theorem. (Integration Bound) For a monotonically increasing function f , � n � n +1 n m − 1 dx f ( x ) ≤ i = m f ( i ) ≤ � dx f ( x ) . m (If f is monotonically decreasing, the inequalities are reversed.) Creator: Malik Magdon-Ismail Sums And Asymptotics: 15 / 16 Integration For Asymptotic Behavior →

Integration For Quickly Getting Asymptotic Behavior Integer Powers. Set f ( x ) = x k : Creator: Malik Magdon-Ismail Sums And Asymptotics: 16 / 16

Integration For Quickly Getting Asymptotic Behavior Integer Powers. Set f ( x ) = x k : n i =1 i k � Creator: Malik Magdon-Ismail Sums And Asymptotics: 16 / 16

Integration For Quickly Getting Asymptotic Behavior Integer Powers. Set f ( x ) = x k : � n n i =1 i k ≈ 0 dx x k � Creator: Malik Magdon-Ismail Sums And Asymptotics: 16 / 16

Integration For Quickly Getting Asymptotic Behavior Integer Powers. Set f ( x ) = x k : 0 dx x k = n k +1 � n n i =1 i k ≈ � k + 1 Creator: Malik Magdon-Ismail Sums And Asymptotics: 16 / 16

Integration For Quickly Getting Asymptotic Behavior Integer Powers. Set f ( x ) = x k : 0 dx x k = n k +1 � n n i =1 i k ≈ k + 1 ∈ Θ( n k +1 ) . � Creator: Malik Magdon-Ismail Sums And Asymptotics: 16 / 16

Integration For Quickly Getting Asymptotic Behavior Integer Powers. Set f ( x ) = x k : 0 dx x k = n k +1 � n n i =1 i k ≈ k + 1 ∈ Θ( n k +1 ) . � Harmonic Numbers. Set f ( x ) = 1 /x (monotonically decreasing): Creator: Malik Magdon-Ismail Sums And Asymptotics: 16 / 16

Integration For Quickly Getting Asymptotic Behavior Integer Powers. Set f ( x ) = x k : 0 dx x k = n k +1 � n n i =1 i k ≈ k + 1 ∈ Θ( n k +1 ) . � Harmonic Numbers. Set f ( x ) = 1 /x (monotonically decreasing): 1 � n +1 n dx 1 H n = � ≤ 1 x i i =1 � �� � ln( n +1) Creator: Malik Magdon-Ismail Sums And Asymptotics: 16 / 16

Integration For Quickly Getting Asymptotic Behavior Integer Powers. Set f ( x ) = x k : 0 dx x k = n k +1 � n n i =1 i k ≈ k + 1 ∈ Θ( n k +1 ) . � Harmonic Numbers. Set f ( x ) = 1 /x (monotonically decreasing): 1 � n +1 � n n dx 1 1 dx 1 H n = � 1 + . ≤ ≤ 1 x x i i =1 � �� � � �� � 1+ln n ln( n +1) Creator: Malik Magdon-Ismail Sums And Asymptotics: 16 / 16

Integration For Quickly Getting Asymptotic Behavior Integer Powers. Set f ( x ) = x k : 0 dx x k = n k +1 � n n i =1 i k ≈ k + 1 ∈ Θ( n k +1 ) . � Harmonic Numbers. Set f ( x ) = 1 /x (monotonically decreasing): 1 � n +1 � n n dx 1 1 dx 1 H n = � 1 + . ≤ ≤ 1 x x i i =1 � �� � � �� � 1+ln n ln( n +1) Stirling’s Approximation for ln n ! . Set f ( x ) = ln x : Creator: Malik Magdon-Ismail Sums And Asymptotics: 16 / 16