The theory of Taylor polynomials and Taylor series rests upon once crucial insight: in order to approximate a function, it is often enough to approximate its value and its derivatives (first, second, third, and so on) at one point. We will see why this is true in the next section; for now, we content ourselves with figuring out how to accomplish it.
Let us first examine polynomials themselves. Setting
|p(x) = a n x n + ... + a 1 x + a 0|
we see that
|p'(x)||=||na n x n-1 + ... +2a 2 x + a 1|
|p''(x)||=||(n)(n - 1)a n x n-2 + ... + (3)(2)a 3 x + 2a 2|
|p (3)(x)||=||(n)(n - 1)(n - 2)a n x n-3 + ... + (4)(3)(2)a 4 x + (3)(2)a 3|
Substituting 0 for x in all of these functions yields
|p(0)||= a 0|
|p(0)||= a 1|
|p(0)||= 2a 2|
|p (3)(0)||= 6a 3|
Indeed, we see a pattern emerging. If we set p (0) = p , then we may write
|p (i)(0) = i!a i|
for i = 0, 1, , n . For i > n , it is easy to see that p (i)(0) = 0 .
Given an arbitrary function f (x) , we want to find a polynomial p(x) such that p (i)(0) = f (i)(0) for, say, i = 0, 1, , n . Since, for a polynomial
|p n(x) = a n x n + ... + a 1 x + a 0|
we have p n (i)(0) = i!a i for i = 0, 1, , n , the coefficients must satisfy
|i!a i = p n (i)(0) = f (i)(0)|
Solving for a i yields
|a i =|
Allowing each a i to take on the value imposed by this equation gives the desired polynomial
|p n(x) = f (0) + f'(0)x + x 2 + x 3 + ... + x n|
called the Taylor polynomial of degree n for the function f (x) .
Now we see the relevance of series. We might guess that as we approximate a function f (x) with higher degree polynomials, having more derivatives in common with f (x) at x = 0 , the resulting polynomials will be better approximations to the actual function f . The natural thing to do is therefore to write down the series that is in some sense the "Taylor polynomial of infinite degree," having the Taylor polynomial of degree n as its n -th partial sum. If we differentiate such a series term-by-term, we see that it will have all of its derivatives at 0 equal to those of f . We let
|p ∞(x) = f (0) + f(0)x + x 2 + x 3 + ... = x n|
This series, called the Taylor series of f at 0 , is a special kind of power series, an object that was explored in the last chapter. As with any power series, a Taylor series may or may not converge for a particular real number x . However, it will converge for all x inside the radius of convergence. In many cases, the Taylor series will define a function that is equal to the original function f (x) inside this radius of convergence.
Sometimes it is convenient to approximate a function using its derivatives at a point other than 0 . For this, consider the function g(x) = f (x + a) . By repeated applications of the chain rule, g (n)(x) = f (n)(x + a) , so the Taylor series for f (x + a) is
|x n = x n|
Letting y = x + a , we have the following Taylor series for f (y) :
|(y - a)n|
This new series is called the Taylor series of f at a . Letting a = 0 , we see that we get back the original Taylor series at 0 .