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: 𝑎 𝑛 =3𝑛−2 𝑎 𝑛 =3+(−1 ) 𝑛

Alternating Sign Find the first four terms: 𝑎 𝑛 = (−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: 𝑎 1 =3 𝑎 𝑘 =2 𝑎 𝑘−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 𝑖=1 𝑛 𝑎 1 + 𝑎 2 + 𝑎 3 +…+ 𝑎 𝑛−1 + 𝑎 𝑛 Plug in for i starting with 1 and finishing with n and add all of the results. Evaluate: 𝑖=1 5 3𝑖 𝑥=0 6 𝑥 2 −1

Series The sum of terms of a sequence The sum of the first n terms is called the nth partial sum 𝑖=1 𝑛 𝑎 𝑖 = 𝑎 1 + 𝑎 2 + 𝑎 3 +…+ 𝑎 𝑛−1 + 𝑎 𝑛 The sum of all terms of a sequence is called an infinite series 𝑖=1 ∞ 𝑎 𝑖 = 𝑎 1 + 𝑎 2 + 𝑎 3 +…

Find the sum Find (a)the third partial sum and (b) the sum of 𝑖=1 ∞ 3 10 𝑖

Homework p. 613: 9, 13, 23, 33-55 odd, 59-69 odd, 70, 79-95 odd