Sequences and Series
In this chapter we’ll be taking a look at sequences and (infinite) series. Actually, this chapter will deal almost exclusively with series. However, we also need to understand some of the basics of sequences in order to properly deal with series. We will therefore, spend a little time on sequences as well.
Series is one of those topics that many students don’t find all that useful. To be honest, many students will never see series outside of their calculus class. However, series do play an important role in the field of ordinary differential equations and without series large portions of the field of partial differential equations would not be possible.
In other words, series is an important topic even if you won’t ever see any of the applications. Most of the applications are beyond the scope of most Calculus courses and tend to occur in classes that many students don’t take. So, as you go through this material keep in mind that these do have applications even if we won’t really be covering many of them in this class.
Here is a list of topics in this chapter.
Sequences We will start the chapter off with a brief discussion of sequences. This section will focus on the basic terminology and convergence of sequences
More on Sequences Here we will take a quick look about monotonic and bounded sequences.
Series The Basics In this section we will discuss some of the basics of infinite series.
Series Convergence/Divergence Most of this chapter will be about the convergence/divergence of a series so we will give the basic ideas and definitions in this section.
Series Special Series We will look at the Geometric Series, Telescoping Series, and Harmonic Series in this section.
Integral Test Using the Integral Test to determine if a series converges or diverges.
Comparison Test/Limit Comparison Test Using the Comparison Test and Limit Comparison Tests to determine if a series converges or diverges.
Alternating Series Test Using the Alternating Series Test to determine if a series converges or diverges.
Absolute Convergence A brief discussion on absolute convergence and how it differs from convergence.
Ratio Test Using the Ratio Test to determine if a series converges or diverges.
Root Test Using the Root Test to determine if a series converges or diverges.
Strategy for Series A set of general guidelines to use when deciding which test to use.
Estimating the Value of a Series Here we will look at estimating the value of an infinite series.
Power Series An introduction to power series and some of the basic concepts.
Power Series and Functions In this section we will start looking at how to find a power series representation of a function.
Taylor Series Here we will discuss how to find the Taylor/Maclaurin Series for a function.
Applications of Series In this section we will take a quick look at a couple of applications of series.
Binomial Series A brief look at binomial series.