MAT01A1. Appendix E: Sigma Notation

Transcription

1 MAT01A1 Appendix E: Sigma Notation Dr Craig 5 February 2019

2 Introduction Who: Dr Craig What: Lecturer & course coordinator for MAT01A1 Where: C-Ring 508 Web:

3 Important information Course code: MAT01A1 NOT: MAT1A1E, MAT1A3E, MATE0A1, MAEB0A1, MAA00A1, MAT00A1, MAFT0A1 Learning Guide: available on Blackboard. Please check Blackboard twice a week. Student check this account twice per week or set up forwarding to an address that you check frequently.

4 Important information Lecture times: Tuesday 08h50 10h25 Wednesdays 17h10 18h45 Lecture venues: C-LES 102, C-LES 103 Tutorials: Tuesday afternoons (only!) 13h50 15h25: D-LES 104 or B-LES 102 OR 15h30 17h05: C-LES 203

5 Important information Textbook: the textbook for this module is Calculus: Early Transcendentals (International Metric Edition) James Stewart 8th edition An e-book version of the textbook is provided to every MAT01A1 student. Click on miebooks on ulink. If you want to buy a hard copy, older editions are fine.

6 Other announcements 2nd year IT students who have a clash with the tutorial on a Tuesday afternoon must Dr Craig today. No MAT01A1 tuts on Wednesdays. If you see this on your timetable, it is an error. Students with Chem. prac clash: Mr Kgatshe ckgatshe@uj.ac.za Need help? Visit the Maths Learning Centre in C-Ring 512.

7 Lecturers Consultation Hours Monday: 14h40 15h25 Dr Craig (C-508) Tuesday: 11h20 13h45 Dr Robinson (C-514) Wednesday: 15h30 17h05 Dr Robinson (C-514) Thursday: 11h20 12h55 Dr Craig (C-508) Friday: 11h20 12h55 Dr Craig (C-508)

8 Sigma notation If a m, a m+1,..., a n are real numbers and m, n Z such that m n, then n a i = a m + a m+1 + a m a n 1 + a n The letter i is called the index of summation. Other letters can also be used as the index of summation. The number of terms in the sum is n m + 1.

9 Examples: (a) 4 i 2 = = 30 (b) n i = (n 1) + n (c) (d) i=3 3 2 j = = 15 j=0 n k=1 1 k = n

10 More examples: (e) (f) 3 i 1 i = = = = = 8

11 Exercise: Write the sum n 3 in sigma notation. Solution(s): Sigma notation for a particular sum is not unique. Some possible solutions: n 3 = n i 3 i= n 3 = n 1 (j + 1) 3 j= n 3 = n 2 (k + 2) 3 k=0

12 Which of the options below are correct sigma notation for the sum: (a) 8 (3 + 2i) (b) (c) (d) 8 2(1 + j) j=1 5 k= 2 [2 + (k + 3)] 7 (3 + 2m) m=0

13 Theorem 2: Let c be any constant. Then n (a) ca i = c n (b) (c) a i n (a i + b i ) = n a i + n (a i b i ) = n a i n b i n b i

14 Theorem 2: Let c be any constant. Then n (a) ca i = c n (b) (c) a i n (a i + b i ) = n a i + n (a i b i ) = n a i n b i n b i Proof: (a) follows from the distributive law: ca m + ca m ca n = c(a m + a m a n ). (b) follows from the commutativity and associativity of addition. (c) combine the last two results.

15 Note that in general ( n n ) ( n ) (a i b i ) a i b i For example: = 2 and 2 = 4 but 2 (1 2) = 4 8.

16 Example: Find n 1. Solution: n 1 = }{{} n terms = n Example: Prove the formula for the sum of the first n positive integers: n n(n + 1) i = 2

17 Telescoping sums: n Consider j=0 ( 1 j j + 2 ). This is equal to ( ) ( ) ( ) ( ) (n 1) (n 1) + 2 ( 1 + n ) = 1 1 n + 2 n + 2

18 Example: Prove the formula for the sum of the squares of the first n positive integers: n i 2 = n 2 n(n + 1)(2n + 1) = 6 Proof: Let S = n i 2. Consider the telescoping sum: n [(1 + i) 3 i 3 ] =...

19 Theorem 3: Let c be a constant and n a positive integer.then (a) n 1 = n (b) n c = nc (c) n (d) n (e) n i = n(n+1) 2 i 2 = n(n+1)(2n+1) 6 i 3 = [ n(n+1) ] 2 2

20 The formulas in the previous theorem can also be proved by Mathematical Induction. We will teach the method of Mathematical Induction in Week 5. To see a method for proving formula (e) with a telescoping sum, see Exercise 39 in the textbook.

21 Which of the options below are correct sigma notation for the sum: (a) n (2 + 5i) (b) (c) 5 [12 + 5(j 2)] j=1 5 (2 + 5m) k=1 (d) 4 [7 + 5(l 1)] l=0

22 Examples: Evaluate Find lim n n i(4i 2 3) n 3 i ) 2 ] + 1 n[( n

23 An important example of sigma notation: Therefore 1 + x1 1! + x2 2! + x3 3! + x4 4! +... = 1 + x + x2 2 + x3 6 + x x n = n! = ex n=0 e = e 1 =

24 Extra exercises (solutions on next slide): Write in expanded form: n+3 j=n Write in sigma notation: j

25 Solutions to extra exercises: Write in expanded form: n+3 j=n j 2 = n 2 + (n + 1) 2 + (n + 2) 2 + (n + 3) 2 Write in sigma notation: ( 1) i (3 i ) OR ( 1) k+1 (3 k 1 ) OR... i=0 k=1