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Calculus IV
15.Functions of Several Variables
16.Multiple Integration
17.Vector Analysis
S = contains supplemental resources
Course: College Algebra
Topic: Sequences and Series
Subtopic: Introduction to Sequences and Series

Overview

Pop quiz! Can you complete these sequences?
a. 5, 11, 17, 23, 29, ___, ___, ___
b. 2, 4, 8, 16, 32, ___, ___, ___
c. 3, 5, 7, 11, 13, ___, ___, ___
d. 1, 1, 2, 3, 5, 8, ___, ___, ___
Think about them before reading on.

Answers:
a. 35, 41, 47. To get the subsequent terms you add 6. This is an example of an arithmetic sequence.
b. 64, 128, 256. To get the subsequent terms you multiply by 2. This is an example of a geometric sequence.
c. 17, 19, 23. This is simply a sequence of prime numbers.
d. 13, 21, 34. Can you see that the subsequent terms are formed by combining previous terms (1+1=2, 1+2=3, 2+3=5, 3+5=8, etc.)? This is an example of a recursive sequence.

This lesson introduces general sequences (lists) and series (sums) -- the notation, terminology, and processes. We'll follow this lesson with a concentration on specific types of sequences and series (arithmetic and geometric). This material will be used particularly in Calculus III when we will write functions such as ex as a sum of infinitely many rational terms (ex = 1 + x + x2/2! + x3/3! + x4/4! + ...). This will enable us to perform calculus on rational expressions rather than on the (more complicated) transcendental function itself.

Objectives

By the end of this topic you should know and be prepared to be tested on:

Terminology

Define: sequence, finite sequence, infinite sequence, series, finite series, infinite series, terms (of a sequence or series), summation, summation notation, sigma notation, upper limit, lower limit, index of summation, sum, partial sum, sequence of partial sums, factorial, alternating sequence, Fibonacci sequence, recursively-defined (a.k.a. recursive) sequence, properties of series

Text Notes

The Fibonacci Sequence is the most famous example of a recursive sequence. If you haven't investigated this very rich topic before or would like to learn more, please explore the links below.

If your text discusses electronically producing "dot" graphs of a sequence you may SKIP these examples/problems throughout the chapter.

Supplemental Resources

Pages to explore the Fibonacci sequence and its connections to Pascal's Triangle and the Golden Mean:
  Biography of Leonardo Fibonacci
  Dr. Knott's Fibonacci Numbers and Nature ~ a must read!
  Pascal's Triangle and its Patterns
  The Golden Ratio/Section/Mean/Number
Want more sites to explore or some books to read? Just ask! This is a favourite topic of mine, practically a hobby :)