An introductory talk about power series with my son and a surprise (to me) misconception

I’m wrapping up sequences and series this week with my son and the final topic is Taylor Series. We’d had a few discussions here and there about power series, but it all comes together this week. Looking through some old problems form the BC calculus test, I found a nice one from 2011 that I wanted to use to introduce the idea of error terms.

I intended for the first three parts to be review, but one interesting misconception came up – so the talk was more than just review.

Here’s the introduction to the problem and my son’s work on the first part of the problem. This problem asks you to write down the usual series for \sin(x) and then write down the series for \sin(x^2)

The next question asks you to write the series for \cos(x) and then write the Taylor Series around x = 0 for the function \cos(x) + \sin(x^2).

Here my son wrote the series for the 2nd function in a way that surprised me:

Once we wrote the correct series for the 2nd part of the last question, we moved on to part (c) of the problem -> find the 6th derivative of the function above evaluated at x = 0:

Finally, we looked at the last part of the problem. The question is about the error in a Taylor series approximation. I’d hoped to use this question to introduce ideas about error terms in Taylor Series, but unfortunately I completely butchered the discussion. Oh well – we’ll be covering the ideas here in a much more detailed way later this week:

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Working through a series problem from the 2012 BC calculus exam

We are about to start the section on power series, but since I haven’t blogged about our calculus work in a while I wanted to do a blog post about testing convergence of series. I chose this problem from the 2012 BC calculus exam:

Screen Shot 2018-12-06 at 7.44.47 PM

Here is my son’s work on the first series:

Here’s his work on the 2nd series:

Here’s his work on the 3rd series:

This has been a fun topic to cover. I’m excited to start on power series tomorrow!

A nice problem about primes for kids from James Tanton

Saw a really cool tweet from James Tanton today:

Tonight I sat down with the boys to make sure they understood the problem. They noticed that half the numbers would have no powers of two – good start! After that observation they started down the path to solving the problem really quickly. In fact, my younger son thought that we might have a geometric series.

Since we covered a few ideas pretty quickly in the last video, so I stared this part of the project by asking them to give me a more detailed explanation for how they got the 1/2, 1/4, 1/8, . . . pattern in the last video. It turned out to be a little harder for them to give precise arguments, but they did manage to hit the main points which was nice.

At the end of this video my older son was able to write down the series that we needed to add up to solve the problem.

Now that we had the series, we had to figure out how to add it up. My guess was that they’d never seen a series like this, but my older son had a really cool idea almost immediately – rewrite the series!

The boys were able to sum the series in this new form – so yay!

At the end of of the last video my younger son said that he was surprised that the “expected value” wasn’t zero since zero was the most likely value. In this part of the video we talked a bit more about what “expected value” meant.

Once we had that I asked what I meant to be a quick question -> is the expected number of 3’s higher or lower. It turned out to be a longer conversation than I expected, though, because my older son was actually able to write down the answer!

Definitely a fun problem. I think it is fun for kids to see how to add up a series like the one in this project. I also think it is fun for kids to explore some of the basic ideas about primes that pop up in this problem.

As an aside, one other place where I’ve seen the series that came up here is in this post from Patrick Honner:

Proof Without Words: Two Dimensional Geometric Series

His “proof without words” for the sum is this picture – can you see how it works?

Mr Honner Square

Sharing Kelsey Houston-Edwards’s Infinity video with kids

The latest PBS Infinite Series video came out this week:

This is the 4th video in an incredible series from Kelsey Houston-Edwards. Our projects on the first 3 are here:

Sharing Kelsey Houston-Edward’s [higher dimensional spheres] video with kids

Sharing Kelsey Houston-Edward’s Philosophy of Math video with kids

Sharing Kelsey Houston-Edward’s Pigeonhole Principle with kids

I had the boys watch the new video together and started today’s project by asking them what they thought was interesting.

After hearing what the kids found interesting, we dove into the idea of bijections. We talked a bit about how a bijection has to work both ways using the bus idea from the video.

After the bus example we moved on to the example of the bijection between the points in an interval and points on the real line.

We finished up by talking about the bijection between the national numbers and the positive even integers.

Since we’ve done several prior projects where infinity played some role, the next thing I asked the kids was for some thing that they already knew about infinity – both things that they thought made sense and things that they thought didn’t make sense. The discussion and examples here were amazing – “no one knows what infinity divided by infinity is” 🙂

Finally, we wrapped up the project talking about why the infinity associated with the real numbers is larger than the infinity associated with the natural numbers.

I thought this would be a fun way to end the project since it was one of the key ideas in Houston-Edwards’s video:

So, another really fun project from the new set of math videos from PBS Infinite Series. I love this new series – can’t wait for the next one!

Talking about infinite series with kids

When we did our “math biographies” project a few weeks ago I asked the kids to tell me about something in math that they heard was true but that they do not believe is true.  My younger son mentioned the seemingly strange property of the Koch Snowflake having an infinite perimeter but a finite area, and my older son brought up this series:

1 + 2 + 3 + . . . .  = -1/12

The “math biographies” project is here:

Math Biographies for my kids

Our two projects about the Koch Snowflake are here:

Exploring the perimeter of the Koch snowflake

Talking through the area of the Koch Snowflake with kids

And a little bit of my thoughts leading up to the project today are here (including the 4 screen shots of passages in G. H. Hardy’s book “Divergent Series”):

Explaining how 1 + 2 + 3 + . . . can possibly equal -1/12 to a kid

I think that explaining to kids how the series 1 + 2 + 3 + . . . can possibly equal -1/12 is practically impossible. You might actually be able to explain the idea to high school calculus students – and it would be fun to try – but, anyway, today’s project doesn’t quite go all the way to an explanation.

Instead what I tried to do was show some other strange results with infinite series that would seem to not have a sum and then walked the boys through some of the philosophical ideas in Hardy’s book.

We started with two infinite series that they were already familiar with:

(1) 1 + 1/2 + 1/4 + 1/8 + . . . = 2, and

(2) 0.99999999….. = 1.

Listening to what both kids have to say about these two series, I’m actually pretty happy with how they understand them.

 

For the second part of the project I extended the idea in the “proof” that the boys used to show that 0.99999…. = 1 and applied it to the series

9 + 90 + 900 + 9000 + . . . .

They were surprised to see that using exactly the same idea that they had just used, this series “sums” to -1.

I loved that my younger son immediately recognized that this strange number:

….99999.999999…. has to equal zero!

 

Now me moved on to looking at geometric series and finding a close formula for the series:

1 + x + x^2 + x^3 + \ldots

Looking at the formula we found, we see (possibly surprising) explanation for the results that we’ve seen previously in this project.

I love the questions that the kids had during this part of the project.

 

Finally we looked at some of the passages in G. H. Hardy’s “Divergent Series” book. I wanted to go through these passages so that (i) the boys could see how a professional mathematician like Hardy thinks about these seemingly strange results, and (ii) more simply, so that the boys could see these strange results in writing.

Some of the ideas are probably a little too philosophical for the boys to understand, but I thought it would still be important for them to see them.

 

So, that’s my best effort 🙂 Maybe several years from now when my older son is learning Calculus, we’ll revisit the topic. It was fun to think about how to share these ideas with kids, and especially nice to hear my younger son say at the end of the last video that this project was “pretty fun.”

Explaining how 1 + 2 + 3 + . . . can possibly equal -1/12 to a kid

When I did the my biographies for my kids last week my older son said that the thing in math that he’s see but that he does not believe is this equality:

1 + 2 + 3 + 4 + . . . . = -1/12

This sum was made popular by a Numberphile video a couple of years ago (which now has over 4 million views!):

 

there have also been several good follow ups. For example this video with Ed Frenkel which was also produced by Numberphile:

 

and this video by Mathologer which is absolutely excellent:

 

I spent some time today trying to think about how to discuss this series with my older son. I’m glad that he is bothered by the result – it is obviously very very strange. Obviously I can’t go into the details about the Riemann Zeta function with him, but I still think there’s some what to help him make some sense of the series. So, I spent the day reviewing some ideas in G. H. Hardy’s book “Divergent Series.” Here are a few passages that caught my eye:

(a) Book Cover

I don’t remember where I heard about this book. My best guess is that it was mentioned in Jordan Ellenberg’s “How Not to be Wrong” in the section about Grandi’s series. Unfortunately I only have the audiobook version of “How not to be Wrong” and don’t know how to search it!

(b) first passage

The remark beginning at “It is plain . . . ” caught my attention.  This is right at the beginning of the book – section 1.3.   The statement:

“it does not occur to a modern mathematician that a collection of mathematical symbols should have a ‘meaning’ until one has been assigned to it by definition.”

also felt very powerful to me.

(c) second passage

The continuation of the previous page is also important – the point about Cauchy was definitely mentioned in “How not to be Wrong” as well.

(d) third passage

 

For the third passage we have to go much later in the book – nearly to the end, in fact.  The passage here – 13.10.11, in particular – shows the strange result.  Not in a Numberphile video, or some other internet video, but in a math textbook by G. H. Hardy:

(e) fourth passage

Finally – and this really is just about the last page of the book – section 13.17 provides a word of caution and an example of what can go wrong playing around with these divergent infinite series.

 

So, I’m going to spend the next few days and maybe even the next few weeks thinking about how to share some sort of idea about this strange series with my son.  I’ll welcome any suggestions!

Talking through the area of the Koch snowflake with kids

This project is the 2nd of two projects on the Koch snowflake. The reason for the projects was that my younger son wondered how the Koch snowflake could have an infinite perimeter but a finite area.

The first project (about the perimeter) is here:

Exploring the perimeter of the Koch snowflake

Our approach to studying the area was similar to the approach for studying the perimeter. Essentially we looked at the steps in the construction of the Koch Snowflake and then looked for a pattern. Here are the initial thoughts from the kids about the area:

 

The first step in studying the area was to look at the total area of the first few iterations of the Koch Snowflake.

I decided to avoid the complexity of geometry triangle formulas and just talked about scaling. My younger son also came up with a really nice argument for

 

Now that we’ve seen a first few cases, can we find the pattern?

The amount of area that we add each time has a fairly simple pattern – it is just multiplication by 4 and division by 9. The only time that doesn’t happen is in the first step.

Can we connect the numbers with the geometry?

 

Now that we’ve seen and understood the pattern, how can we figure out the sum? I love that the boys saw that the main sum we were looking at here was less than 2.

I didn’t want to derive the geometric sum formula, so I just gave it to them. We can talk about it another time. That formula seems to be the easiest way to find the exact value of the sum, though.

 

Finally we wrapped up and discussed the process we used to study the area and perimeter. I don’t really believe that my younger son now understands every detail of what we talked about, but I hope that he’s a little bit less confused about the area and perimeter of the Koch snowflake.

I think the math here is something that all kids would find interesting.