# Summer courses almost over, "children's" books in the works.

This summer is one I will remember. I taught both differential equations and calculus II for the first time. Seeing these subjects from the instructors side has really opened my eyes to all kinds of details I never noticed before. One of the most striking new insights is how much these two courses have in common. They both rely deeply on sequences and series. Sequences are like a hallway in mathematics, one that connects many many many rooms.

I am working on two math book projects. The first is a Japanese-styled art book on the topic of sine and cosine. It's inspired by many of the lessons I taught this summer.

Japanese-style book about sine and cosine

I want to bring all of the different ways that sine and cosine are presented in elementary and undergraduate mathematics in to one (long) pictorial document. I start with the differential equation, $y''+y=0$ then solved it (using the series method from differential equations) producing $\sin x = \sum^{\infty}_{n=0} \frac{(-1)^n}{(2n+1)!} x^{2n+1}$ and $\cos x = \sum^{\infty}_{n=0} \frac{(-1)^n}{(2n)!} x^{2n}$

Next I wanted a pictorial way to relate these power series to the unit circle. I have found it in this spiral (the first image shows how it is constructed as an involution):

Constructing The Involute Pinwheel

a sequence of involutes: the vertical and horizontal components will form the power series for sine and cosine respectively.

The vertical and horizontal components will form the power series for sine and cosine respectively. Take the series of vertical line segments: $\sin x = A_1A_2 - A_3A_4 + A_5A_6 - \cdots$ and so on, the segments repeatedly over and under-shoot the accutal value of sine. The full paper by Leo S. Gurin, "A problem", can be found here.

I'm going to incorporate Gurin's spiral in to my book. I want to show the power series literally flying out of it, like they have come to life. I wonder if I can make it like the famous drawing of the sine curve projecting out of the unit circle?

Naturally, I already have planned to put that diagram in my booklet.

Work in progress

The Japanese-style book is perfect for series and periodic functions It's one long continuous piece of paper:

Yet very compact:

I'm also working on a very silly book about hypercycloids (that's the "math" name for the shapes drawn by spirographs, did I mention I collect spirographs?):

I'm trying to make it like a children's book, fun, light, a little silly:

I can't wait to share the final product.

# Almost Convergence.

The title of my last post "Almost Cauchy Sequence" made me wonder if "Almost Cauchy" was already defined in some formal sense. I could not find much on "Almost Cauchy" ... but I did find out about "Almost Convergence." (I think I have ran in to this before... but it is very neat... observe:)

A sequence $S = S_n$ is almost convergent to L if for any $\epsilon > 0$ we can find an integer n such that the average of n or more consecutive terms in the sequence is within $\epsilon$ of L. Formally, $\forall \epsilon > 0 \quad \exists N \ni \quad \displaystyle \left| \frac{1}{n} \sum_{i=0}^{n-1} S_{k-i} -L \right| < \epsilon \quad \forall n > N, \; k \in \mathbb{N}.$ (I took this definition from here, and I wonder why they didn't mention that we need $k \geq n$ or else we are looking at negative terms in the sequence... But, maybe there is an interpretation of something like $S_{-3}$? Or maybe the author thinks that would be obvious? I'm going to ask at the mathematics stack exchange!)