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The Taylor Series

Approximating Functions With Polynomials



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 .

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