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
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.
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.
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.
Test Using the Ratio Test to determine if a series
converges or diverges.
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.