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 .