11 1 Sequences Sequence A sequence is a

11. 1 Sequences

Sequence A sequence is a list of numbers written in an explicit order. nth term Any real-valued function with domain a subset of the positive integers is a sequence. If the domain is finite, then the sequence is a finite sequence. In calculus, we will mostly be concerned with infinite sequences.

Examples The last example is a recursively defined sequence known as the Fibonacci Sequence.

Limit and Convergence • Let’s take a look at the sequence • What will happen as n gets large? • If a sequence {an} approach a number L as n approaches infinity, we will write and say that the sequence converges to L. • If the limit of a sequence does not exist, then the sequence diverges.

Example Does converge? The sequence converges to 2. Graph the sequence.

Properties of Limits • Same as limit laws for functions in chapter 2. • Theorem: Let f (x) be a function of a real variable such that If {an} is a sequence such that f (n) = an for every positive integer n, then • Squeeze Theorem • Absolute Value Theorem: For the sequence {an},

Examples Determine the convergence of the following sequences.

Monotonic Sequence • A sequence is called increasing if for all n. • A sequence is called decreasing if for all n. • It is called monotonic if it is either increasing or decreasing.

Bounded Sequence • A sequence is bounded above if there is a number M such that an ≤ M for all n. • A sequence is bounded below if there is a number N such that N ≤ an for all n. • A sequence is a bounded sequence if it is bounded above and below. Theorem: Every bounded monotonic sequence is convergent.

Examples Determine whether the sequence is bounded, monotonic and convergent.

A geometric sequence is a sequence in which the ratios between two consecutive terms are the same. That same ratio is called the common ratio. Example: Geometric sequences can be defined recursively: or explicitly:

A sequence is defined recursively if there is a formula that relates an to previous terms. Example: We find each term by looking at the term or terms before it: