Sequences and Series Section 8.1

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.

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.

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

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, . . . }

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

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

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, …

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!

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+1 c) 𝑎 𝑛 = (−1) 2𝑛 2𝑛 !

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!

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 + …

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

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

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

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 ∞ 4 1 10 𝑖

What is the difference between a sequence and a series?

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

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−1 2 + (𝑖+1) 3 ] Find the fifth partial sum: 𝑖=1 ∞ 5 1 2 𝑖 Find the sixth partial sum: 𝑘=1 ∞ 1 10 𝑘 Write an formula for the general of the sequence: 0, 3, 8, 15, 24, …