In this post I introduce the function *f(x)* and look at some of its properties.

**Table of Contents**

1. Outline

2. A polynomial and its derivatives

3. A new function

4. The integral is an integer (pending)

5. The size of the integral (pending)

I’ll go ahead and be a little blunt. Let’s define *f(x)* as

Alright, so what is *n*? That turns out not to matter, as long as it’s a postive integer. We will prove several things about *f(x)* that don’t depend on *n*; later on, we’ll see that the contradiction we are looking for appears only when *n* is “large enough,” but this won’t be a problem.

Note that *f(x)* is symmetric about the line :

This symmetry will come into play later on.

**Some derivatives**

We now show that *f*^{(i)}(0) and *f*^{(i)}(π) are always integers. We first note that the numerator of *f(x)* can be written as

a polynomial of degree *n* through 2*n* whose coefficients are all integers.

*f*^{(i)}(0)

Note that the first *n*-1 derivatives of *f(x)* have no constant term, so those derivatives are all 0, which is an integer. The same goes for the 2*n*+1^{th} and all further derivatives–once we take the 2*n*+1^{th} derivative of a plynomial of degree 2*n* (such as *f(x)*), we have 0. So all we need to think about now are the *n*+1^{th} through 2*n*^{th} derivatives.

As we noted above, *f(x)* is a sum of terms of the form

where and *c*_{i} is an integer. Taking the *m*^{th} derivative reduces this to

This term only becomes significant when we’ve taken the *n+i*^{th} derivative. The term then becomes

which is clearly an integer.

*f*^{(i)}(π)

Note that

since *f(x)* is symmetric about the line , so

We can repeat the process as many times as we want. Thus, the even derivatives of *f(x)* are symmetrix about the line . Since, as we have shown, *f*^{(i)}(0) is always an integer, *f*^{(i)}(π) is always an integer as well.

### Like this:

Like Loading...

*Related*

This entry was posted on 8 June 2010 at 16:49 and is filed under Math, Pi. You can follow any responses to this entry through the RSS 2.0 feed.
You can leave a response, or trackback from your own site.

## Leave a Reply