1. Sequences and Series
Section 8.1

2. Objective:
You will know how to write the terms of a sequence, find the nth term of a sequence, compute and simplify factorials and expressions involving factorials, use summation notation to write the terms of a sequence, and find sums of infinite series.

3. Vocabulary
Infinite sequence: function whose domain is the set of positive integers.
Terms: function values are the terms of the sequence.
If the domain of the function consists of the first n positive integers, then the sequence is a finite sequence. Otherwise, it is an infinite sequence.

4. What is a Sequence?
A sequence is a set of things (usually numbers) that are in order and are separated by commas.

5. Infinite vs. Finite Sequences
If a sequence goes on forever, it is an infinite sequence (the “. . . ” means that the sequence goes on forever).
If a sequence has a first term and a last term, then it is a finite sequence.
Example: Classify the following sequences as either finite or infinite.
{1, 2, 3, 4 ,...} e) {1, 2, 4, 8, 16, 32, ...}
{20, 25, 30, 35, ...} f) {a, b, c, d, e}
{1, 3, 5, 7} g) {0, 1, 0, 1, 0, 1, ...}
{4, 3, 2, 1} h) {f, r, e, d, }

6. Notation: 𝑎 𝑛 𝑎 𝑛 is the term and n is the term number.
Example: 𝑎 1 , 𝑎 2 , 𝑎 3, , 𝑎 𝑛
Example: {3, 5, 7, 9, …}
1) Identify (a) first term, (b) second term, (c) fourth term
2) Identify 𝑎 𝑛 when a)n=5, (b) n=9, (c) n=11, (d) n=13

7. Writing the Terms of a Sequence
Example: Write the first four terms of the following sequences: For example, to find 𝑎 1 , plug in n=1 a) 𝑎 𝑛 =2𝑛+5 b) 𝑏 𝑛 = 3 𝑛−1 c) 𝑐 𝑛 = (−1) 𝑛 𝑛 2 +1

8. Finding the nth term
Write a formula for the nth term of the sequence (i.e. find a formula that relates the term of the sequence to the term number)
1, 3, 5, 7, …
2, 5, 10, 17, …

9. Factorial Notation
Define 𝑛!=𝑛 𝑛−1 𝑛−2 … 2 1 , where 𝑛 is a positive integer.
Special case: 0! = 1
Evaluate the following factorials: 4! 3! 5!
d) 6!

10. Terms of a Sequence Involving Factorials
Write the first four terms of the following sequences. (i.e. Find 𝑎 1 , 𝑎 2 , 𝑎 3 , 𝑎 4 )
a) 𝑎 𝑛 = 𝑛 2 𝑛! b) 𝑎 𝑛 = 𝑛! 2𝑛 c) 𝑎 𝑛 = (−1) 2𝑛 2𝑛 !

11. Evaluating Factorial Expressions
Simplify each factorial expression.
Step 1: Write the factorials as product of numbers, but do not multiply the numbers.
Step 2: Cancel out the numbers that are common in the numerator and the denominator.
a) 5! 7! b) 8! 2! ∙ 6! c) 2! ∙6! 3! ∙5!

12. Series Series: the sum of the terms of a sequence.
Finite series/partial sum: sum of the first n terms of a sequence
𝑎 1 + 𝑎 2 + 𝑎 3 + …+ 𝑎 𝑛
Infinite series: sum of the terms of an infinite sequence
𝑎 1 + 𝑎 2 + 𝑎 3 + …

13. Summation Notation 𝑖=1 𝑛 𝑎 𝑖 = 𝑎 1 + 𝑎 2 + 𝑎 3 + …+ 𝑎 𝑛
𝑖=1 𝑛 𝑎 𝑖 = 𝑎 1 + 𝑎 2 + 𝑎 3 + …+ 𝑎 𝑛
n = upper limit of summation
1 = lower limit of summation
i = index of summation

14. Summation Notation
Write out the first four terms of the following series. Example: Find the following sums: a) 𝑖=1 4 3𝑖 b) 𝑖=1 4 (2𝑖+1) c) 𝑖=1 4 2 𝑖

15. Properties of Sums 𝑖=1 𝑛 𝑐 =𝑐𝑛 𝑖=1 𝑛 𝑐 𝑎 𝑖 =𝑐 𝑖=1 𝑛 𝑎 𝑖
𝑖=1 𝑛 𝑐 𝑎 𝑖 =𝑐 𝑖=1 𝑛 𝑎 𝑖
𝑖=1 𝑛 ( 𝑎 𝑖 + 𝑏 𝑖 ) = 𝑖=1 𝑛 𝑎 𝑖 + 𝑖=1 𝑛 𝑏 𝑖
𝑖=1 𝑛 ( 𝑎 𝑖 − 𝑏 𝑖 ) = 𝑖=1 𝑛 𝑎 𝑖 - 𝑖=1 𝑛 𝑏 𝑖

16. Find the fifth partial sum of 𝑖=1 ∞ 2 1 3 𝑖
Find the sum of the first three terms of the series: 𝑖=1 ∞ 𝑖

17. What is the difference between a sequence and a series?

18. Homework
Page 587, # 1, 3, 11, 21, 24, 42, 47, 51, 56, 62, 72, 87, 107

19. Find 𝑎 16 : 𝑎 𝑛 = (−1) 𝑛−1 [𝑛(𝑛−1)]
Write the first five terms of the sequence: 𝑎 𝑛 = (−1) 𝑛 𝑛 𝑛+1
Write the first five terms of the sequence and find the fourth partial sum: 𝑎 𝑛 = 𝑛(𝑛−1)(𝑛−2)
Write the general term of the sequence: 3, 7, 11, 15, 19, …
Write the first five terms of the sequence: 𝑎 𝑛 = (−1) 2𝑛+1 2𝑛+1 !
7) Simplify the factorial expression: 12! 4!∗8!
Find the sum: 𝑖=1 5 [ 𝑖− (𝑖+1) 3 ]
Find the fifth partial sum: 𝑖=1 ∞ 𝑖
Find the sixth partial sum: 𝑘=1 ∞ 𝑘
Write an formula for the general of the sequence: 0, 3, 8, 15, 24, …