5.2 Infinite Series

Textbook PDF pp. 450–459

Textbook HW: p. 466: 67–74, 76, 77, 79, 80, 83–85 odd, 89—95 odd

Webwork: Intro to Series due 4/19

Notes from lecture and office hours: Note Apr 13, 2020

Video from lecture and office hours https://www.dropbox.com/s/dimiyy58j0p36tn/Lecture%20%26%20Office%20Hours-20200413%201207-1.mp4?dl=0

Motivation

Remember that our ultimate goal this chapter is to understand what it means for an “infinite degree Taylor polynomial” to “represent” a function near a point. We said that our first step toward this is to understand what it means for a sequence to converge, how to determine if a sequence converges, and (if a sequence does converge) determine what it converges to.

Our next step is to understand what it means for a series to converge. The next few sections will cover different tests to determine whether different series converge or diverge.

Introduction

If a sequence is a list of numbers: $a_1, a_2, a_3, \dots$ then a series  is just the sum of the terms in the series: $a_1+a_2+a_3+ \dots$. Sequences and series don’t have to have infinitely many terms, but the ones we’re interested in will. (Language note: the plural of “series” is “series.”)

A good first example of a series to have in mind is one that we talked about in class before we transitioned to online classes (Xeno’s paradox): We saw then that $\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+ \dots$ converges to 1. We talked about this series when we were first talking about improper integrals. Both convergent series and convergent improper integrals are examples of things that are infinite in one way but finite in another, so they kind of mess with our intuition.

Here is a Desmos graph which shows that if we think of a sequence as a discrete versions of a function, then the corresponding series is like a discrete version of definite integral (area under a curve). I should point out in the videos below you’ll see graphs of partial sums; those graphs and the graph linked here are related but not exactly the same.

If you need it, here is a refresher on Sigma notation (the first 4 minutes) before continuing.

Here is an introduction (8 minutes) to some theory of series: the sequence of partial sums of a series and what it means for a series to converge. This video also describes the divergence test for series and shows some motivating examples of some series which converge or diverge. Officially, the divergence test isn’t covered until Section 5.3, but there is one question on the Webwork set Intro to Series that might require it.

Geometric Series

The series above for Xeno’s paradox is an example of a geometric series (which you might remember from your precalculus class). Geometric series are your friends! They’re among the nicest types of series you might hope to meet. One reason that they’re so nice is that if a series is geometric, it’s easy to tell if it converges or diverges. If a geometric series converges, it’s also easy to see what it converges to.

Over the next few classes, we’ll be discussing several different convergence tests for series and some of them might look complicated at first, but I’m going to point out when a particular test is saying, “Well…this series isn’t actually geometric, but it’s practically eventually almost geometric, and that’s good enough.”

Here’s a video (10 minutes) that tells you everything you need to know about infinite geometric series.

Telescoping Series

Telescoping series are another nice type of series. Think of an old-fashioned telescope like this one. If you are a person who needs to carry around a telescope, it’s not practical to carry a big long tube around with you, so you might want one that collapses into a smaller tube (really a bunch of tubes nested inside one another). Telescoping series work much the same way. When you write out the individual terms of the summation, it looks like some big long thing, but if you arrange the terms differently, you’ll see a lot of cancellation, which is like the collapsing of a telescope.

Here’s a video (7 minutes) on telescoping series.

Sum of the first $n$ integers

One formula that you might need to know is the sum of the first $n$ integers:

$\sum_{i=1}^n i= 1 + 2 + \dots + (n-1) + n = \frac{n(n+1)}{2}$.