# 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 .

## Take a Study Break

### Star Trek gets SEXY

Chris Pine and Zoe Saldana heat up the red carpet!

### Are you afraid of relationships?

Auntie SparkNotes can help!

### Sexy starlet style

See every single look from the Met Gala!

### Geeky Actors: Then and Now

Travel back in time!

### Villains We Want These Actresses to Play

From super cute to super bad!

### 10 Movies Better Than Their Books

What do you think?

### How To Look Like J-Law...

When you don't look like J-Law.

### 12 Scientific Inaccuracies in Into Darkness

What did Star Trek get wrong?

## The Book

### Read What You Love, Anywhere You Like

Get Our FREE NOOK Reading Apps