Aside — Fourier

The whole purpose of this is to have the form of a function that is easy to manipulate, in terms of their drivatives.

So what Fourier did was find a way to represent a function as an infinite sum of sines and cosines, because their derivatives are really nice.

So if we can prove we can represent the same function as a sum of sines and cosines.

Something something, key equation problem

Taylor Series

If you have a continuous and differentiable function f at x_0, we can rewrite f(x) as

f(x) = \sum_{n=0}^{\infin} \frac{ f^{(n)} (x_0) (x - x_0)^n }{ n! }

Example: A very easy example.

Q:

f(x) = 3x^2 + 4x - 7, x_0=3

TODO Something something the Taylor series wraps around ???

For easy reference, f'(x) = 6x+4

A:

T_f^{(1)} (x) = f(3) = 32

T_f^{(2)} (x) = 32 + f'(3)(x-3) = 22x - 34

Now, let’s find T_f^{(3.1)} (x)

T_f^{(3.1)} (x) = 34.2

TODO I don’t get this dawg.

Let’s go back.

T_f^{(3)} (x) = 22x - 34 + 3(x-3)^2 = 3x^2 + 4x - 7

If the function is a polynomial, it’ll take n terms (where n is the degree of the function) for the Taylor series to converge.

This example is pretty rote, so let’s go onto something a little more interesting

Example:

f(x) = ln(x) + 1 , x_0 = 3

A:

T_f^{(1)} (x) = f(3) = 2.09861228866810969140

T_f^{(2)} (x) = 2.09861228866810969140 + \frac{1}{3} (x-3)

Let’s try getting T_f^{(3.1)} (x).

f(3.1) = 2.13140211149110056191

T_f^{(3.1)} (x) = 2.131945622

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Since the function is more interesting now, let’s try 3.5

f(3.5) = 2.13140211149110056191

T_f^{(3.5)} (x) = 2.241469432

If we keep going, we’ll diverge because ???

TODO These values are just the delta addd to the initial point, that’s pretty cool (what did he mean by this)

T_f^{(3)} (x) = 2.09861228866810969140 + \frac{1}{3} (x-3) + (- \frac{1}{18}) (x-3)^2

Let’s evaluate this function at 3.1

T_f^{(3.1)} (x) = 2.131390067

T_f^{(3.5)} (x) = 2.25139400

I think he’s trying to show it’s more accurate. This seems self-evident, but also, it is not self-evident, LMFAO. TODO

Let’s be bold, let’s get 4.

T_f^{(4)} (x) = 2.376390067

We will never converge, no matter how many terms, but ??? TODO.