MTH 253 Calculus (Other Topics) Chapter 11 – Infinite Sequences and Series Section 11.1 – Sequences Copyright © 2009 by Ron Wallace, all rights reserved.

Primary Goal of this Chapter How are values of transcendental functions (and others) determined or approximated? transcendental = non-algebraic Algebraic Functions All functions that can be expressed using a finite number of elementary operations (, , , , and roots); along with their inverses. Transcendental Functions include, Trigonometric & inverses Hyperbolic & inverses Exponential & inverses (i.e. logarithms) Other Uses of Infinite Series: solve DE’s, difficult Integrals, model physical laws Express in terms of algebraic functions?

Secondary Goal of this Chapter How do you add up an infinite number of numbers?

Sequence Formally: A function whose domain is a set of integers. Intuitively: An ordered list of numbers (called terms). The Domain: Usually, the positive integers (can start anywhere). Can be finite, however this chapter is only concerned with infinite sequences and therefore “sequence” will imply “infinite sequence.”

Sequence A function whose domain is the set of integers [n,  ). Notation: Meaning: Example: n can start anywhere and it is not required that a n has a formula General Term

Sequence A function whose domain is the set of integers [n,  ). How to determine the general term from a list of terms. No simple well defined process. Pattern recognition. Check your hypothesis. Useful Tips (assume n starts at 1) : integers starting at k = n+(k-1) evens = 2n odds = 2n-1 alternating signs (1 st positive): (-1) n+1 alternating signs (1 st negative): (-1) n

Sequences Examples: Finding the general term from a list of terms.

Sequences One More Example Find the first three terms of the sequence... What does this imply?ODD Integers! Careful, don’t jump to conclusions to quickly!

Graphing Sequences Domain? Integers in the interval [a,). Graph? Disconnected set of points (i.e. not continuous) Related continuous Function? Replace n with x and use all points in the interival [a,) for the domain.

Limit of a Sequence Given: Find: With sequences n  Z + and a n is not continuous, therefore the limit definition must be modified. Limit definition for x   and f(x) a continuous function.

Limit of a Sequence Given: Find: Limit definition for n   and { a n } a sequence. L L+ L– 1234N ●●●

Limit Theorems for Sequences Just like functions! Also, all methods of limit evaluation for functions, applies to sequences. Converges  limit exists Diverges  limit does not exist NOTE: Alternating sequences can only converge to 0. Why?

Defining Sequences Recursively Each term is determined in terms of one or more previous terms. OR … whatever … Fibonacci Sequence:

Some Common Limits … x > 0 |x| < 1 Note that with these limits, x remains fixed (i.e. a relative constant).

Does the sequence converge? What a sequence converges to is not always important, but does it converge to something? Can you determine if a sequence converges without finding the actual limit.

Terminology How would you describe ?

Is increasing or decreasing? Two Tests Difference Between TermsRatio of Successive Terms (use both methods)

A Third Test using Derivatives Is increasing or decreasing?

Properties that Hold Eventually If a finite number of the terms from the beginning of a sequence are discarded and the resulting sequence has a property, then the original sequence has that property eventually. Example What can be said about: The sequence is eventually increasing. (discard the first 4 terms)

Convergence of Sequences If a sequence is eventually increasing or non-decreasing, then there are two possibilities: 1. M (upper bound) where all a n  M and the sequence converges to a value L  M. 2.No upper bound and the sequence approaches infinity as n. Note: In the first case, finding M guarantees convergence without the need to find L. L = “least upper bound”

Convergence of Sequences If a sequence is eventually decreasing or non-increasing, then there are two possibilities: 1. M (lower bound) where all a n  M and the sequence converges to a value L  M. 2.No lower bound and the sequence approaches negative infinity as n. Note: In the first case, finding M guarantees convergence without the need to find L. L = “greatest lower bound”

Convergence of Sequences Examples Eventually decreasing and always positive. (why?) Therefore it converges. (why?) Eventually increasing but no upper bound. (why?) Therefore it diverges. (why?)