Study Guide

Series

Series Introduction

Around 490 BC, there was a philosopher named Zeno who didn't believe in motion. Just to be clear, he didn't believe that things move. We're as confused as you are. Philosophers have been debating his arguments for the last two thousand years.

Although Zeno had lots of arguments against the existence of motion, there's one in particular that usually shows up in calculus classes under the name Zeno's Paradox. Here is what this madman thought up:

Suppose you want to eat a double fudge brownie on the other side of the room. In order to reach the brownie, first you have to get halfway there. Then you have to go half the remaining distance: Then half the remaining distance again. No matter how many times you move, you will never get taste the chocolate goodness . That sounds like something Zeno would come up with, especially if the brownie doesn't move.

You're probably thinking this is totally ridiculous. If you saw a brownie on the other side of the room you could walk over and eat it, no philosophy required. You know that you can walk across the room, but Zeno says you can't, what's going on?

Zeno's paradox breaks up the distance to the brownie into an infinite sum of finite numbers, also known as a series. Say your initial distance from the brownie is 1 unit. At every step, you will always travel half the distance you have left to the brownie. So your first step is  unit, then you second step is  unit, and so on.

The total distance you need to travel is

The problem is that we (and Zeno) don't know what this means. We don't really know what is means to add infinitely many numbers. Do we ever get to the brownie 1 unit away? Leave it to Zeno to leave us salivating over a square of fudge.

This final chapter of calculus is all about adding infinitely many numbers. We are going to learn what a series is, and then we will learn how to add infinitely many numbers using convergence and divergence of series. Once we've enlightened ourselves, we can add up all the distances in the brownie situation and resolve the paradox.

Series Resources

Websites

Series Cheatsheet
This isn’t a list of your favorite TV shows. This is a list of the most common infinite series that show up in mathematics, along with the convergence tests. But if you find the Golden Girls in this list, don’t be surprised.

Review List of Series Convergence Tests
Having trouble finding the basket-weaving test for series convergence? There’s a good reason why. This is an encyclopedic link for series convergence tests.

Solved Series Convergence Problems
Just because you have a jack-hammer doesn’t mean you know how to use it. Learn how to use the different series convergence tests from these solved examples.

Videos

Testing a Geometric Series for Convergence or Divergence
Like the travelers on the Oregon Trail, some geometric series will get where they are going. Others will never make it, left to wander aimlessly. This video will show you how to tell those who will make it from those that will not.

Integral Test Example
Integrals: the gifts that keep on giving. This link shows you how to use them to solve series convergence problems.

Alternating Series Test Examples
It’s not an electrical socket, but you still can’t put your fork in it. Learn how to use the alternating series test to determine if the series is convergent.

Conditional Convergence, Absolute Convergence, and Divergence Examples
With so many different types of convergence, you could use a video guide for different series that fit the descriptions. Wait until you learn there are different types of infinity too.

Ratio Test Example
KISS: Keep It Simple, Silly. The ratio test is easily the simplest of all the series tests. It works well, most of the time. Learn how here.

Comparison Test Example
When you come across an animal you don't recognize in the woods, how do you know if it is going to eat you or let you live? You compare it to other animals. With the comparison test, you can do the same with infinite series.

Games and Tools

Strategy Guide on Solving Series Convergence Problems
You don’t need to be a psychic to figure out which series test is the best to use. This is a simple strategy guide that will act a trailhead so you can find your way.