(This material was originally posted for class on Wednesday, March 25 but was moved to Monday, March 30 and then moved again to Monday, April 6)

Here are the notes from this lesson and office hours: Note Apr 6, 2020

Here is the video:

https://www.dropbox.com/s/1xc9k8owuq4jn27/Lecture%20%26%20Office%20Hours-20200406%201255-1.mp4?dl=0

5.1 Sequences

Textbook PDF p.427–444

Homework: (p.427–444) P. 447: 1, 3, 7, 9, 12, 13–15 odd, 23–37 odd, 47–51 odd

Webwork: Sequences due 3/31

Motivation

In the Taylor polynomials section, we saw that as the degrees of the Taylor polynomials of $f(x)$ centered at $x=a$ increased, the better they approximated the original function $f(x)$ near $x=a$. We said something like, “the best approximation would be a polynomial of infinite degree.” The thing is: a polynomial of infinite degree doesn’t exist…but we call those things “power series.” So if a polynomial is the finite sum of a bunch of powers of $x$ (with coefficients), then a power series is the infinite sum of a bunch of powers of $x$ (with coefficients).

Before we can ask about what it means to have the sum of infinitely many powers of $x$ (a power series), we’re first going to ask what it means to have the sum of infinitely numbers (a “series”). But actually! First we’re going to ask what it means to have just a list of infinitely many numbers (a “sequence”) instead of a sum.

Introduction

If you took Precalculus at CityTech (MAT 1375), you’ve already seen an introduction to sequences (and series), so this will be mostly review for you.

  • Here are a review video (8 minutes).
  • tl;dr: a sequence is just a list of numbers.
Convergence

The thing we’ll be most interested in about a sequence is whether it converges.

Something that the above video doesn’t really exploit enough is that a sequence is just a function where the domain is all positive or non-negative integers. For example, the sequence $a_n =\frac{1}{n}$ is the same thing as the function $f(x) =\frac{1}{x}$ as long as $x$ is a positive integer. I can graph the function $f(x)$ (in orange) and the sequence (in blue) together; the sequence consists of $y$-values where $x$-values are the positive integers (link to graph).

We say that a sequence $\{a_n\}$ converges if the limit $\lim_{n \to \infty}$ exists. (That is, the limit is an actual finite number.) If we think about the corresponding function $f(x)$, this just means that $f(x)$ has a horizontal asymptote as $x$ approaches $\infty$. It doesn’t matter what the sequence is doing for small values of $n$, but it does matter what it’s doing for large values of $n$. So we are only really interested in the behavior of the sequence eventually.

In the above example, the sequence $a_n = \frac{1}{n}$ converges to the value 0 because $\lim_{n \to \infty} \frac{1}{n} = 0$….but we already used that $\lim_{x \to \infty} \frac{1}{x} = 0$ to show that the function $f(x) = \frac{1}{x}$ has a horizontal asymptote (as $x$ approaches $+ \infty$) of $y=0$.

Boundedness, monotonicity (increasing or decreasing)

Some features of a sequence (or its corresponding function) can help you determine whether it converges or diverges. A sequence/function is called monotone if it is either increasing or decreasing (so its graph doesn’t bounce up and down). A sequence/function is called bounded if it always stays between two values (graphically, its graph stays between two horizontal lines).

It’s helpful to know: if a sequence is bounded and monotonic, then it converges.