9.1 Sequences

A sequence is a list of numbers written in an explicit order. n th 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. Sequence

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

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

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. Geometric sequences can be defined recursively: Example: or explicitly:

If the second term of a geometric sequence is 6 and the fifth term is -48, find an explicit rule for the nth term. Example

Let’s take a look at the sequence What will happen as n gets large? If a sequence {a n } 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. Limit and Convergence

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

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

Examples Determine the convergence of the following sequences.

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. Monotonic Sequence

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

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