## Why π is irrational (part 3 of 5)

In this post I use the derivatives of f(x) to create a new function, F(x), that will help us evaluate the integral from part 1.

1. Outline
2. A polynomial and its derivatives
3. A new function
4. The integral is an integer
5. The size of the integral (pending)

We define the new function as the alternating sum of the even derivatives of f(x). In symbols,



Using sigma notation, we can write this more compactly as



First, note that F(0) and F(π) are both integers. We have already determined that all derivatives of f(x) at 0 and π are integers, so any sum or difference of those derivatives must also be an integer. F(0) and F(π) are such sums or differences; thus, they must be integers.
Now consider F”(x)+F(x). Note from the definition of F that



The final term should techinically become



but as we have noted previously, this is zero, so we ignore it. Note that every term in F(x) has the opposite sign of the corresponding term of F”(x), with the exception of f(x), which does not have a corresponding term in F”(x). Thus,