Sequences and Series
9.1
Sequence
• A set of numbers, determined by some function
infinite if the domain of that function is all positive integers
finite if the domain of that function consists of the first n
integers.
Writing the Terms of a Sequence
Write the first four terms of the sequence given by:
𝑎𝑛 = 3𝑛 − 2
𝑎𝑛 = 3 + (−1)𝑛
Alternating Sign
• Find the first four terms:
𝑎𝑛 =
(−1)𝑛
2𝑛+1
Find the Pattern
• Find the nth term of the sequence:
1,3,5,7, …
2, −5, 10, −17, …
Recursive Sequence
Write the first four terms of the sequence defined by:
𝑎1 = 3
𝑎𝑘 = 2𝑎𝑘−1 + 1, 𝑘 ≥ 2
Fibonacci Sequence
Write a recursive rule for the Fibonacci Sequence
1, 1, 2, 3, 5, 8, 13, …
Factorial
n! = 1 x 2 x 3 x 4 x … x (n-1) x n
Summation Notation
𝑛
𝑖=1 𝑎1
+ 𝑎2 + 𝑎3 + ⋯ + 𝑎𝑛−1 + 𝑎𝑛
Plug in for i starting with 1 and finishing with n and add all of
the results.
Evaluate:
5
𝑖=1 3𝑖
6
2
𝑥
𝑥=0
−1
Series
• The sum of terms of a sequence
• The sum of the first n terms is called the nth partial sum
𝑛
𝑎𝑖 = 𝑎1 + 𝑎2 + 𝑎3 + ⋯ + 𝑎𝑛−1 + 𝑎𝑛
𝑖=1
• The sum of all terms of a sequence is called an infinite series
∞
𝑎𝑖 = 𝑎1 + 𝑎2 + 𝑎3 + ⋯
𝑖=1
Find the sum
• Find (a)the third partial sum and (b) the sum of
∞
3
10𝑖
𝑖=1
Homework
• p. 613: 9, 13, 23, 33-55 odd, 59-69 odd, 70, 79-95 odd