Algebra I / Elem. Algebra
Introduction to Algebra Linear Equations and Inequalities Functions and Graphs I Lines and thier Graphs Linear Systems
Algebra II / E&I Algebra
Exponents & Polynomials Intermediate Algebra starts here!

Factoring Rational Expressions Rational Equations and Applications
Algebra III / Inter. Algebra
Radical Expressions Nonlinear Equations and Applications Functions and Graphs II Exponential and Logarithmic Functions
College Algebra
Equations and Inequalitites Functions and Graphs Polynomial and Rational Functions Exponential and Logarithmic Functions Systems and Matrices Geometry Basics Conic Sections Sequences and Series
Trigonometry
The Six Trigonometric Functions Right Triangle Trigonometry Circular Functions Graphs of Trigonometric Functions Trigonometric Identities Trigonometric Equations Oblique Triangles and the Laws Vectors Complex, Parametric, and Polar Forms
Calculus I
Limits and Continuity Derivatives Analysis of Curves Antiderivatives
Calculus II
Transcendental Functions
Geometry Physics Integration Techniques Calculus of Infinity Parametric, Polar, and Conic Curves
Calculus III
Calculus IV
15.Functions of Several Variables
16.Multiple Integration
17.Vector Analysis

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

Overview

A series is a sum of terms where the terms have some pattern connecting them. Series can be finite such "the sum of the first five prime numbers" 2+3+5+7+11, or infinite such as "the series of positive odd numbers" 1+3+5+7+... going on forever and ever. There are many reasons one might need to write a series to analyze a problem such as studying a bouncing ball going down then up then down then up but not quite as far and adding the total distance traveled by the ball during its bounces. Really pretty fun stuff when you get into it, but a fair amount of terminology and formulae involved too.

Objectives

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

• 11.2.1 Produce examples of series, both finite and infinite, and have an intuitive understanding of their having a finite sum or not
• 11.2.2 Evaluate or simplify a series algebraically and electronically
• 11.2.3 Given a series, write an explicit formula for the n-th term
• 11.2.4 Write a series, both finite and infinite, using summation notation
• 11.2.5 Convert a series written in summation notation to a new sum with a different lower or upper index
• 11.2.6 Write the sequence of partial sums of an infinite series
• 11.2.7 Determine if the sequence of partial sums converges and how it applies to the value of the series
• 11.2.8 Apply the properties of series as needed

Terminology

Define: series, infinite series, summation notation, index of the summation, explicit formula, n-th term, sequence of partial sums, convergent series, divergent series

Supplemental Resources (recommended)

Be sure that you can evaluate a summation electronically (eg. using Wolfram Alpha), not just algebraically. For TI graphing calculators see Keely's Calculator Guide: Sequences and Series.

Supplemental Resources (optional)

Paul's OL Notes - Calc II: Series: The Basics

Selwyn Hollis's Video Calculus: Infinite Series

Patrick JMT Just Math Tutorials:
Summation Notation
What is a Series?
Showing a Series Diverges using Partial Sums

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.