1. Sequences & Series MATH 109 - Precalculus S. Rook

2. Overview Section 9.1 in the textbook: – Infinite sequences – Factorial notation – Partial sums & summation notation 2

3. Infinite Sequences

4. 4 We have discussed finite (countable) lists of numbers when constructing a table of values: – Given a function f(x), pick values of x to get f(x) – We do this about 2 or 3 times to get an idea what f(x) looks like – Represents only a subset of the values of f(x) – i.e. a Finite Sequence Infinite Sequence: a function whose domain is the natural numbers. The results that are generated from a sequence are its terms There are many infinite sequences of interest to mathematicians and scientists – Prime numbers, Fibonacci numbers, etc.

5. 5 Terms of a Sequence The n th term of a sequence also called the general term is usually written a n = f(n) Given a natural number k such that 1 ≤ k ≤ n, we can find the k th term of the sequence by simply substituting – i.e. a k = f(k)

6. 6 Alternating Sequences Alternating Sequence: a sequence in which subsequent terms change from positive to negative or vice versa – Has a general term such as a n = (-1) n + 1 · f(n) Substitute as before to evaluate a term

7. Infinite Sequences (Example) Ex 1: For each sequence, find the first three terms and then the 10 th term: a) b) 7

8. 8 Recursive Sequences Recursive Sequence: a sequence defined in terms of itself using previous terms – Usually given at least the first term of the sequence – e.g. a n + 1 = 5 + a n ; a 1 = 2

9. Recursive Sequences (Example) Ex 2: Find the first three terms of the recursive sequence: a) b) 9

10. Factorial Notation

11. 11 Factorial Notation Suppose we were give the recursive sequence a n = n · a n – 1 ; a 1 = 1 n = 2: a 2 = 2 · a 1 = 2 · 1 = 2 n = 3: a 3 = 3 · a 2 = 3 · (2 · 1) = 3 · 2 = 6 n = 4: a 4 = 4 · a 3 = 4 · (3 · 2 · 1) = 4 · 6 = 24 : : a n = n · (n – 1) · (n – 2) · … · 2 · 1

12. 12 Factorial Notation (Continued) a n = n · (n – 1) · (n – 2) · … · 2 · 1 is used often enough that it is given the special name factorial and written as n! n! means the product of n down to 1 3! = 3 · 2! = 3 · 2 · 1! = 3 · 2 · 1 = 6 1! AND 0! are both equivalent to 1 n! = n · (n – 1)! We can use factorials when performing Algebraic operations – By expanding the factorial into a product

13. Factorial Notation (Example) Ex 3: Evaluate the factorials by hand: a) b) c) 13

14. Partial Sums & Summation Notation

15. 15 Partial Sums We have seen how to generate successive terms from the sequence a n = f(n) Another important series concept is the summation of these terms The summation through the n th term is called the n th partial sum denoted S n S 1 = a 1 S 2 = a 2 + a 1 S 3 = a 3 + a 2 + a 1 : S n = a n + a n-1 + … + a 2 + a 1 Each of the n th partial sums forms a sequence S n is also called a finite series

16. Summation Notation A shorthand way to write the partial sum from the m th term to the n th term where m ≤ n is where ∑ means to sum the elements from m to n of the sequence a n m is known as the lower limit (starting value) of the summation (does not always have to start at 1) n is known as the upper limit (ending value) of the summation k (in this case) is known as the index of summation (other variables can be used as well) 16

17. Summation Notation (Continued) – The summation of ALL terms of an infinite sequence is known as an infinite series denoted in summation notation as 17

18. Summation Notation (Example) Ex 4: Evaluate: a)b) c)d) 18

19. Summary After studying these slides, you should be able to: – Calculate the terms of the following types of sequences: Infinite Alternating Recursive – Understand factorial notation and be able to perform simple calculations – Evaluate partial sums and series of a sequence using summation notation Additional Practice – See the list of suggested problems for 9.1 Next lesson – Arithmetic Sequences & Partial Sums (Section 9.2) 19