9.1 Sequences Sequences are ordered lists generated by a function, for example f(n) = 100n

9.1 Sequences A sequence is a function that has a set of natural numbers as its domain. f (x) notation is not used for sequences. Write Sequences are written as ordered lists a1 is the first element, a2 the second element, and so on

9.1 Sequences A sequence is often specified by giving a formula for the general term or nth term, an. Example Find the first four terms for the sequence Solution

9.1 Graphing Sequences The graph of a sequence, an, is the graph of the discrete points (n, an) for n = 1, 2, 3, … Example Graph the sequence an = 2n. Solution

9.1 Sequences A finite sequence has domain the finite set {1, 2, 3, …, n} for some natural number n. Example 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 An infinite sequence has domain {1, 2, 3, …}, the set of all natural numbers. Example 1, 2, 4, 8, 16, 32, …

9.1 Convergent and Divergent Sequences A convergent sequence is one whose terms get closer and closer to a some real number. The sequence is said to converge to that number. A sequence that is not convergent is said to be divergent.

9.1 Convergent and Divergent Sequences Example The sequence converges to 0. The terms of the sequence 1, 0.5, 0.33.., 0.25, … grow smaller and smaller approaching 0. This can be seen graphically.

9.1 Convergent and Divergent Sequences Example The sequence is divergent. The terms grow large without bound 1, 4, 9, 16, 25, 36, 49, 64, … and do not approach any one number.

9.1 Sequences and Recursion Formulas A recursion formula or recursive definition defines a sequence by Specifying the first few terms of the sequence Using a formula to specify subsequent terms in terms of preceding terms. (Using the previous term to plug into the formula)

9.1 Using a Recursion Formula Example Find the first four terms of the sequence a1 = 4; for n>1, an = 2an-1 + 1 Solution We know a1 = 4. Since an = 2an-1 + 1

9.1 Applications of Sequences Example The winter moth population in thousands per acre in year n, is modeled by for n > 2 Give a table of values for n = 1, 2, 3, …, 10 Graph the sequence.

9.1 Applications of Sequences Solution (a) (b) n 1 2 3 4 5 6 an 2.66 6.24 10.4 9.11 10.2 7 8 9 10 9.31 10.1 9.43 9.98 Note the population stabilizes near a value of 9.7 thousand insects per acre.

9.1 Series and Summation Notation A finite series is an expression of the form and an infinite series is an expression of the form .

9.1 Series and Summation Notation Sn is the sum a1 + a2 + …+ an of the first n terms of the sequence a1, a2, a3, … .  is the Greek letter sigma and indicates a sum. The sigma notation means add the terms ai beginning with the 1st term and ending with the nth term. i is called the index of summation.

9.1 Series and Summation Notation Example Evaluate (a) (b) Solution (a) (b)

9.1 Series and Summation Notation Summation Properties If a1, a2, a3, …, an and b1, b2, b3, …, bn are two sequences, and c is a constant, then for every positive integer n, (a) (b) (c)

9.1 Series and Summation Notation Summation Rules

9.1 Series and Summation Notation Example Use the summation properties to evaluate (a) (b) (c) Solution (a)

9.1 Series and Summation Notation Solution (b) (c)