# Derivation of Taylor Series

1. Apr 9, 2005

### amcavoy

I understand what a linear approximation, and how it is derived using the point-slope formula:

$$f(x)\approx f(a)+f'(a)(x-a)$$

These are the first three terms of a Taylor series, so I was wondering how the rest was derived?

Thanks for your help.

2. Apr 9, 2005

### Hurkyl

Staff Emeritus
Here's one way of thinking of it:

Can you write down a polynomial p(x) such that:
p(a) = f(a)
p'(a) = f'(a)
p''(a) = f''(a)
?

What about another polynomial q(x) such that:
q(a) = f(a)
q'(a) = f'(a)
q''(a) = f''(a)
q'''(a) = f'''(a)
?

The intuition here is that if the first few derivatives of two functions agree at a, then the two functions should look similar near a.

There's probably some nifty way of measuring the goodness of an approximation of a function near a point by a polynomial, and the taylor polynomials will be the best ones. Maybe it's something like this:

$$\lim_{h \rightarrow 0} \int_{a-h}^{a+h} (f(t) - p(t))^d \, dt$$

for your favorite integer d.

Last edited: Apr 9, 2005