Taylor Series Approximations

A Taylor series can be though of as a Taylor polynomial that goes on forever. It will be examined at which points such a series serves as a good approximation to a function.

Following is a sequence of five Taylor polynomials centered at x = 0 for the function cos x:

cos x ≈ P0(x) = 1
cos x ≈ P0(x) = 1 − 
x2
2!
cos x ≈ P0(x) = 1 − 
x2
2!
 + 
x4
4!
cos x ≈ P0(x) = 1 − 
x2
2!
 + 
x4
4!
 − 
x6
6!
cos x ≈ P0(x) = 1 − 
x2
2!
 + 
x4
4!
 − 
x6
6!
 + 
x8
8!

The above sequence of polynomials is:

P0(x), P2(x), P4(x), P6(x), P8(x), ⋅⋅⋅,

each of which is a better approximation to cos x than the previous, at least for x near 0. Notice that when going to a higher degree polynomial (say from P4 to P6), more terms are added (x6/6!, for example), but the terms of the lower degree don't change. Thus each polynomial includes information from all the previous ones. The Taylor series for cos x is then written:

T (x) = 1 − 
x2
2!
 + 
x4
4!
 − 
x6
6!
 + 
x8
8!
  − ⋅⋅⋅

This represents the whole sequence of Taylor polynomials.


Taylor Series for sin x, cos x, and e x

The Taylor series for sin x and e x can be defined in a similar way to the Taylor series for cos x:

sin (x) = x − 
x3
3!
 + 
x5
5!
 − 
x7
7!
 + 
x9
9!
  − ⋅⋅⋅

cos (x) = 1 − 
x2
2!
 + 
x4
4!
 − 
x6
6!
 + 
x8
8!
  − ⋅⋅⋅

e x = 1 + x + 
x2
2!
 + 
x4
4!
 + 
x6
6!
 + 
x8
8!
  − ⋅⋅⋅

These series are also called Taylor expansions of the functions sin x, cos x, and e x about x = 0.

Taylor Series in General

Any function f, all of whose derivatives exist at 0, has a Taylor series. However, for many functions, f , the series does not converge to f (x) for all values of x. Assuming that the series does converge to f (x), the following formula holds:

Taylor series for f (x) about x = 0

 f (x) =  f (0) + f ' (0)x +
 f '' (0)
2!
x2 + 
 f ''' (0)
3!
x3 + 
 f (4) (0)
4!
x4 + ⋅⋅⋅ + 
 f n (0)
n!
xn + ⋅⋅⋅

In addition, just as Taylor polynomials can be centered at points other than 0, a Taylor series can be centered at x = a (provided all the derivatives exist at x = a). Assuming the series converges to f (x), the following equation results:

Taylor series for f (x) about a = 0

 f (x) =  f (a) + f ' (a)(x − a) +
 f '' (a)
2!
(x − a)2 + 
 f ''' (a)
3!
(x − a)3 + 
 f (4) (a)
4!
(x − a)4 + ⋅⋅⋅ + 
 f n (a)
n!
(x − a)n + ⋅⋅⋅

For many functions f , the Taylor series converges to f (x) only for x near a.

⇐ Prev Lesson: Taylor Polynomials