Learning Resources
Outcomes
In this lesson you will learn
- the concept of "limit"
- the difference between a converging sequence and a diverging sequence
By the end of this section students will be able to:
- demonstrate an intuitive understanding for the concept of limit
- investigate and apply the concept of infinity by examining sequences and series
- represent a series in expanded form and using sigma notation
- develop, analyze and apply algorithms to determine the sum of a series
- demonstrate an understanding of convergence and divergence
- apply convergent and divergent geometric series
- demonstrate an understanding of how to approximate the area under a curve using limits
Introduction
This section is designed as a means of introducing you to the definition of a "limit" in the context of sequences. Basically you will address the concept of limit of a sequence.
You will be introduced to two new important concepts, converging and diverging sequences, the understanding of which will prove to be important as you attempt to use limits to define sums of converging infinite geometric series.
As well, you will use your newly gained knowledge of limits to discover techniques for calculating areas of regions under curves.
Did you ever wonder what a particular repeating decimal was as a fraction? Well, you will use infinite geometric series to calculate the fractional equivalent of repeating decimals. In other words, you will be able to examine a repeating decimal such as 0.345345345... and precisely write the fraction equivalent.
Something to think about: Two people are arguing over whether
is the same as 1 or if it just a little less than 1. What do you think? Hopefully by the end of this section, you will have a clearer view about who is right here.
Zeno, a Greek philosopher once argued that it is impossible for a person standing in a room to walk to the wall. Sound weird? His argument was based on the notion that to get to the wall, the person would first have to go half the distance, and then go half the remaining distance. He could continue in this manner forever. You will have an opportunity to decide whether or not you agree with Zeno's paradox in the Challenge Yourself part of this section.
This section should take approximately 6 hours to complete.
Prerequisites
To be successful in this lesson, it would be helpful to know the following:
- how to readily perform computations that involve fractions
- perimeter and area formulas for simple polygons
- properties of the Fibonacci sequence
- how to use the TRACE function of the graphing calculator