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Infinite Geometric Series 2815 Views


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Description:

Help Pepe on his quest to once again become the "Pizza King." Or put that healthy metabolism to use and steal his title. Ten slices in less than a minute? That's nothing.

Language:
English Language

Transcript

00:04

Infinite Geometric Series, a la Shmoop. Pepe holds the world record as the fastest [Pepe with his trophy in front of lots of pizza boxes]

00:08

pizza eating human.

00:10

Back in 2010 he inhaled 10 slices in 60 seconds.

00:14

He was crowned the “Pizza King” and has held the record ever since.

00:18

But in the years since, he has let himself go.

00:21

He can barely muster up the motivation to make his way to the phone to order one.

00:24

Still, he finds a way.

00:26

One day he is sitting in a pizza joint with some friends when he is challenged to a pizza [Pepe sat with his friends and a pizza is chucked onto the table]

00:31

eating contest by an unknown stranger.

00:33

Pepe wants to relive the glory days, but he isn’t sure he has what it takes.

00:37

Just to be safe, he first eats half a pizza.

00:41

Then one-sixth, one-eighteenth, one-fifty-fourth, and so on...

00:43

If he keeps eating the pizza this way forever, how much pizza will he have eaten in total?

00:52

Well, first let's look at the numbers we have. [Boy looking through binoculars]

00:54

One-half, one- sixth, one-eighteenth, one-fifty-fourth...and so on...

00:58

…looks like we have something called a series.

01:01

Or actually, since we're assuming he'll eat forever... an infinite series, [Pizza slowly disappearing and a clock ticking]

01:06

But let's take a closer look at the numbers and we notice a pattern.

01:11

Between the first two terms, one-half and one-sixth, there's a ratio of one-third.

01:17

Between one-sixth and one-eighteenth, we have the same ratio of one-third.

01:22

Between one-eighteenth and one-fiftieth fourth, the same ratio, one-third.

01:26

We can call this number one-third, the common ratio, or r, of the series. [Old women answers the phone]

01:33

Because the terms of the series are separated by a constant ratio, we can describe the series

01:37

even more specifically, as an infinite geometric series.

01:42

Now that we've identified the type of numbers we're dealing with and the common ratio between [Fraction locked behind bars]

01:46

them, let's get back to the problem.

01:48

If we want to find the total amount of pizza Pepe ate, we should add all the slices together

01:53

and find the sum.

01:54

But wait, if he's eating infinite slices, how can we find a finite sum?

01:59

No worries, Pepe… the sum of an infinite geometric series has a finite sum as long

02:05

as the absolute value of the common ratio is less than one.

02:09

In this case, the absolute value of one-third is one-third, which is less than one. [Teacher at the front of class writing on a whiteboard]

02:15

As you can see from the series above, the numbers are getting smaller and smaller…

02:19

…one-fifty-fourths would go to one over 162, which is getting closer and closer to 0.

02:26

So as you add smaller and smaller numbers, the addition of such small numbers doesn't

02:31

matter much.

02:32

Ok, so let's find that total for Pepe. [Addition of fractions formula]

02:34

The formula for the sum of an infinite geometric series equals the first term divided by the

02:40

quantity one minus the common ratio.

02:44

Substituting our pizza values into the formula, we see that the sum will equal one half divided

02:48

by the quantity one minus one third.

02:51

One minus one-third equals two-thirds, so we're left with one half divided by two thirds.

02:56

Instead of dividing one half by that nasty fraction two thirds, let’s multiply by its

03:01

reciprocal three over two, to get one half times three halves, or 3-fourths.

03:09

That’ll be three quarters of a pizza.

03:12

You can do that, right Pepe? [Pepe picks up the 3/4 of pizza]

03:14

We have faith in you. Here's the water, and the Pepto-bismol and the barf bag... [Arms hold out water etc.. for Pepe]

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