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Calculus III

Course: Calculus III
Topic: Sequences and Infinite Series
Subtopic: Introduction to Sequences and Series

Overview

This lesson introduces general sequences (lists) and series (sums) -- the notation, terminology, and processes -- including specific types of sequences and series (e.g., arithmetic and geometric). Sequences are basically lists of numbers that have some pattern. For example {2, 5, 8, 11, 14, ...} is a sequence starting at 2 and increasing each time by 3. Series involve adding the terms of a sequence together, for example -2+5-8+11-14+17-20+... is an alternating (sign) infinite series.

You may have studied sequences and series in a precalculus math course, but let's take this opportunity to review before more formally studying the calculus of sequences and infinite series over the next few lessons. One goal in Calculus III is to write functions such as e^x as a sum of infinitely many rational terms (e^x = 1 + x + x^2/(2!) + x^3/(3!) + x^4/(4!) + ...). This will enable us to perform calculus on rational expressions rather than on the (more complicated) transcendental function itself.

Try me: 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, ___, ___, ___
To check your answers, join the discussion in class. HINT: one is a recursive sequence, one is an arithmetic sequence, and one is a geometric sequence. Can you write a second one of each?

Objectives

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

• 11.0.1 Evaluate or simplify a factorial algebraically and electronically
• 11.0.2 Write the first few terms of a sequence given the an term
• 11.0.3 Write the next few terms of a sequence given the first few terms
• 11.0.4 Understand alternating sequences and recursive sequences
• 11.0.5 Know the difference between sequences and series
• 11.0.6 Evaluate or simplify a series algebraically and electronically
• 11.0.7 Use proper mathematical notation and format when working with factorials, sequences, and series
• 11.0.8 Apply properties of series as needed

Terminology

Terms you should be able to 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, arithmetic sequence, geometric sequence, Fibonacci sequence, recursively-defined sequence, sequence of partial sums, 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 "Dr. Knott" site linked below.

Your text includes "dot" graphs of sequences that enable visual representations for pattern recognition. Some handheld calculations have a "sequence" option in the grapher. While it is not required for you to be able to produce these graphs electronically, they can be fun to explore, especially if you enjoy playing with technology.

Supplemental Resources (recommended)

Download/Print Prof. Keely's Sequences and Series Formula Sheet (from precalculus)

While not calculus based, pick at least one of these sites to learn more about the Fibonacci sequence and its connections to nature, Pascal's Triangle, and the Golden Mean:
Biography of Leonardo Fibonacci
Dr. Knott's Fibonacci Numbers and Nature and Part 2 ~ both full of interesting info
Pascal's Triangle and its Patterns
The Golden Ratio/Section/Mean/Number

Supplemental Resources (optional)

Paul's OL Notes - Calc II: Sequences and Series - Basics

Selwyn Hollis's Video Calculus: Sequences 1

More tutorial videos if you need them are listed at James Sousa's MathIsPower4U - Calc II. Scroll down middle column to "Infinite Series" for related titles.