How Is the Full Taylor Series Derived Beyond Its Linear Approximation?

[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.

 

[Hurkyl]

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.

 

[M98Ranger]

The Taylor series is a mathematical representation of a function as an infinite sum of terms that are calculated from the values of the function and its derivatives at a specific point. It is named after the mathematician Brook Taylor who first described the concept in the 18th century.

To derive the full Taylor series, we start by expanding the linear approximation formula that you mentioned:

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

We can rewrite this as:

f(x)=f(a)+f'(a)(x-a)+R_1(x)

where R_1(x) is the remainder term, which represents the difference between the actual function and its linear approximation.

Now, we can further expand the remainder term by using the second derivative of the function:

R_1(x)=\frac{f''(a)}{2!}(x-a)^2+R_2(x)

where R_2(x) is the second-order remainder term.

Continuing this process, we can expand the remainder term to include higher-order derivatives of the function:

R_n(x)=\frac{f^{(n)}(a)}{n!}(x-a)^n+R_{n+1}(x)

where R_{n+1}(x) is the (n+1)-th order remainder term.

Substituting these expanded remainder terms into our original equation, we get:

f(x)=f(a)+f'(a)(x-a)+\frac{f''(a)}{2!}(x-a)^2+\frac{f^{(3)}(a)}{3!}(x-a)^3+...+\frac{f^{(n)}(a)}{n!}(x-a)^n+R_{n+1}(x)

This is the general form of the Taylor series, which includes an infinite number of terms. However, in practice, we usually only use a finite number of terms to approximate a function.

To summarize, the Taylor series is derived by expanding the linear approximation formula and including higher-order derivatives of the function in the remainder term. This allows us to approximate a function with increasing accuracy as we include more terms in the series.