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Series

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 1/2 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.

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