Chapter 4

1. Chapter 4 Sequences and Mathematical Induction

2. 4.1 Sequences

3. Sequences • The main mathematical structure used to study repeated processes is the sequence. • The main mathematical tool used to verify conjectures about patterns governing the arrangement of terms in sequences is mathematical induction.

4. Example • Ancestor counting with a sequence • two parents, four grandparents, eight great-grandparents, etc. • Number of ancestors can be represented as 2position • Example: 23 = 8 (great grandparents), therefore parents removed three generations are great grandparents for which you have a total of 8.

5. Sequences • Sequence is a set of elements written in a row as illustrated on prior slide. (NOTE: a sequence can be written differently) • Each element of the sequence is a term. • Example • am, am+1, am+2, am+3, …, an • terms a sub m, a sub m+1, a sub m+2, etc. • m is subscript of initial term • n is subscript of final term

6. Example • Finding terms of a sequence given explicit formulas • ak = k/(k+1) for all integers k ≥ 1 • bi = (i-1)/i for all integers i ≥ 2a • the sequences a and b have the same terms and hence, are identical

7. Example • Alternating Sequence • cj = (-1)j for all integers j≥0 • sequence has bound values for the term. • term ∈ {-1, 1}

8. Example • Find an explicit formula to fit given initial terms • sequence = 1, -1/4, 1/9, -1/16, 1/25, -1/36, … • What can we observe about this sequence? • alternate in sign • numerator is always 1 • denominator is a square • ak = ±1 / k2 (from the previous example we know how to create oscillating sign sequence, odd negative and even positive. • ak = (-1)k+1 / k2

9. Summation Notation • Summation notation is used to create a compact form for summation sequences governed by a formula. • the sequence is governed by k which has lower limit (1) and a upper limit of n. • This sequence is finite because it is bounded on the lower and upper limits.

10. Example • Computing summations • a1 = -2, a2 = -1, a3 = 0, a4 = 1, and a5 =2.

11. Example • Computing summation from sum form.

12. Example • Changing from Summation Notation to Expanded form

13. Example • Changing from expanded to summation form. • Find a close form for the following:

14. Separating Off a Final Term • A final term can be removed from the summation form as follows. • Example of use: Rewrite the following separating the final term

15. Example • Combining final term

16. Telescoping Sum • Telescoping sum can be evaluated to a closed form.

17. Product Notation Recursive form

18. Example • Compute the following products:

19. Factorial • Factorial is for each positive integer n, the quantity n factorial denoted n! is defined to be the product of all the integers from 1 to n: • n! = n * (n-1) *…*3*2*1 • Zero factorial denoted 0! is equal to 1.

20. Example • Computing Factorials

21. Properties of Summations and Products • Theorem 4.1.1 • If am, am+1, am+2, … and bm, bm+1, bm+2, … are sequences of real numbers and c is any real number, then the following equations hold for any integer n≥m:

22. Examples • Let ak = k +1 and bk = k – 1 for all integers k

23. Examples • Let ak = k +1 and bk = k – 1 for all integers k

24. Transforming a Sum by Change of Variable • Transform the following by changing the variable. • summation: • change of variable: j = k+1 • Solution: • compute the new limits: • lower: j=k+1, j=0+1=1 • upper: j=k+1, j=6+1=7