Taylor series

Taylor series

Contents:

1

Real Taylor series

2

Taylor polynomials

3

Examples

4

Complex Taylor series

5

Taylor series and polynomials in Banach spaces

6

Taylor series and polynomials for functions of several variables

7

Taylor expansion of formal polynomials

1

Real Taylor series

Let

f

:

I

→

ℝ

be a

function

defined on an

open interval

I

,
possessing

derivatives

of all orders at

a

∈

I

.
Then the

power series

T

⁢

(

x

)

=

∑

k

=

0

∞

f

(

k

)

⁢

(

a

)

k

!

⁢

(

x

-

a

)

k

is called the

Taylor series

for

f

centered at

a

.

Often the case

a

=

0

is considered, and we have the simpler

T

⁢

(

x

)

=

∑

k

=

0

∞

f

(

k

)

⁢

(

0

)

k

!

⁢

x

k

,

called the

Maclaurin series

for

f

by some authors.

If we perform formal term-by-term

differentiation

of

T

⁢

(

x

)

,
we find that

T

(

k

)

⁢

(

a

)

=

f

(

k

)

⁢

(

a

)

, so it is plausible
that

T

is an extrapolation or an approximation to

f

based
on the derivatives of

f

at a single point

a

.

In general,

T

may not extrapolate or approximate

f

in the strictest sense:
a Taylor series does not necessarily

converge

, and even if it does,
it may not necessarily converge to the original function

f

(

http://planetmath.org/InfinitelyDifferentiableFunctionThatIsNotAnalytic

). It is also not necessarily true that a Taylor series about

a

equals the Taylor series of

f

about some other point

b

,
when considered as functions.

Those functions whose Taylor series do converge
to the function are termed

analytic

(

http://planetmath.org/Analytic

) functions

.

If we start with a

convergent

power series

f

⁢

(

x

)

=

c

0

+

c

1

⁢

(

x

-

a

)

+

c

2

⁢

(

x

-

a

)

2

+

⋯

to
define a function

f

, then the Taylor series of

f

about

a

will turn out to be the same as our original power series.

2

Taylor polynomials

The

n

th degree

Taylor polynomial

for

f

centered at

a

is the

polynomial

P

n

,

a

⁢

(

x

)

=

∑

k

=

0

n

f

(

k

)

⁢

(

a

)

k

!

⁢

(

x

-

a

)

k

.

In general,

P

n

,

a

has degree

≤

n

;
it may be

<

n

if some of the terms

f

(

k

)

⁢

(

a

)

vanish. Nevertheless,

P

n

,

a

is characterized by the following properties:
it is the unique polynomial of degree

≤

n

whose derivatives up to the

n

th order at

a

agree
with those of

f

;
it is also the unique polynomial

p

of degree

≤

n

such that

f

⁢

(

x

)

-

p

⁢

(

x

)

=

o

⁢

(

|

x

-

a

|

n

)

,

as

x

→

a

.

(

Landau notation

is being used here.)
These characterizations are sometimes helpful in
actually computing Taylor polynomials.

The Taylor polynomial

P

n

,

a

is applicable even if

f

is only

differentiable

n

times at

a

, or when its Taylor series does
not converge to

f

.

The error from the approximation of

f

by

P

n

,

a

,
or

remainder term

,

R

n

,

a

⁢

(

x

)

=

f

⁢

(

x

)

-

P

n

,

a

⁢

(

x

)

,
can be quantified precisely
using Taylor’s theorem (

http://planetmath.org/TaylorsTheorem

).
In particular, Taylor’s Theorem is often used to show

lim

n

→

∞

⁡

R

n

,

a

⁢

(

x

)

=

0

,

which is equivalent to

T

⁢

(

x

)

=

f

⁢

(

x

)

, i.e. the Taylor series
converges to the original function.

A term that is often heard is that of a “

Taylor expansion

”;
depending on the circumstance, this may mean either the Taylor series
or the

n

th degree Taylor polynomial.
Both are useful to linearize or otherwise reduce the analytical complexity of a function. They are also useful for numerical approximation of functions,
when the magnitude of the later terms fall off rapidly.

3

Examples

Using the above definition of a Taylor series about

0

, we have the following important series representations:

e

x

=

1

+

x

1

!

+

x

2

2

!

+

x

3

3

!

+

⋯

sin

⁡

x

=

x

1

!

-

x

3

3

!

+

x

5

5

!

-

x

7

7

!

+

⋯

cos

⁡

x

=

1

-

x

2

2

!

+

x

4

4

!

-

x

6

6

!

+

⋯

That the series on the right converge to the functions on the left
can be proven by Taylor’s Theorem.

4

Complex Taylor series

If

f

:

U

→

ℂ

is a

holomorphic function

from an open subset

U

of the

complex plane

, and

a

∈

U

, we may also consider its Taylor series about

a

(defined with the same formulae as before, but with

complex numbers

).

In contrast with the complex case, it turns out that all holomorphic functions
are infinitely differentiable and have Taylor series that converge to them.
(The

radius of convergence

of the Taylor series at

a

being the radius of the largest

open disk

about

a

on which the domain of

f

can be extended.)

This of course makes the theory of

analytic functions

very nice, and
many questions about real power series and

real analytic functions

are more easily
answered by looking at the complex case.
For example, we can immediately tell that
the Taylor series about the origin for the

tangent function

(

http://planetmath.org/DefinitionsInTrigonometry

)

sin

⁡

z

cos

⁡

z

=

tan

⁡

z

=

z

+

1

3

⁢

z

3

+

2

15

⁢

z

5

+

17

315

⁢

z

7

+

62

2835

⁢

z

9

+

⋯

has a radius of convergence of

π

/

2

,
because

tan

is holomorphic everywhere except at its poles at

z

=

π

/

2

+

k

⁢

π

,

k

∈

ℤ

,
and

π

/

2

is the

distance

that the closest of these poles get to the origin.

5

Taylor series and polynomials in Banach spaces

Taylor series and polynomials can be generalized to Banach spaces:
for details, see Taylor’s formula in Banach spaces.

6

Taylor series and polynomials for functions of several variables

The simplest Banach spaces are the spaces

ℝ

n

,
and in this case Taylor series and Taylor polynomials for functions

f

:

ℝ

n

→

ℝ

(“functions of

n

variables”)
look like this:

T

⁢

(

x

)

=

∑

i

1

=

0

∞

⋯

⁢

∑

i

n

=

0

∞

f

(

i

1

,

i

2

,

…

,

i

n

)

⁢

(

0

)

i

1

!

⁢

i

2

!

⁢

⋯

⁢

i

n

!

⁢

x

1

i

1

⁢

x

2

i

2

⁢

⋯

⁢

x

n

i

n

,

f

(

i

1

,

i

2

,

…

,

i

n

)

⁢

(

0

)

=

∂

i

1

+

⋯

+

i

n

⁡

f

∂

⁡

x

1

i

1

⁢

⋯

⁢

∂

⁡

x

n

i

n

|

x

=

0

,

P

N

,

0

⁢

(

x

)

=

∑

i

1

+

⋯

+

i

n

≤

N

f

(

i

1

,

i

2

,

…

,

i

n

)

⁢

(

0

)

i

1

!

⁢

i

2

!

⁢

⋯

⁢

i

n

!

⁢

x

1

i

1

⁢

x

2

i

2

⁢

⋯

⁢

x

n

i

n

=

∑

|

I

|

≤

N

1

I

!

⁢

∂

|

I

|

⁡

f

∂

⁡

x

I

|

x

=

0

⁢

x

I

.

(For simplicity, we have put the centre at

a

=

0

.
The last expression employs a commonly-used multi-index notation.)

For example, the second-degree Taylor polynomial for

f

⁢

(

x

,

y

)

=

cos

⁡

(

x

+

y

)

centered about

(

0

,

0

)

is

P

2

,

0

⁢

(

x

,

y

)

=

1

-

1

2

⁢

x

2

-

x

⁢

y

-

1

2

⁢

y

2

.

Note that

P

2

,

0

⁢

(

x

,

y

)

can also be obtained by
taking the one-variable Taylor series

cos

⁡

t

=

1

-

t

2

/

2

+

⋯

and substituting

t

=

x

+

y

, and keeping only the terms of degree

≤

2

.
This procedure works because of the uniqueness characterization of Taylor polynomials.

7

Taylor expansion of formal polynomials

If

f

is a polynomial function, of degree

n

,
then its Taylor series and its Taylor polynomial of degree

≥

n

actually equal

f

. For this reason, we can consider Taylor series and polynomials
applied to formal polynomials, without any notion of

convergence

.
(The usual derivative is replaced by formal differentiation.)
In this setting, a “Taylor expansion” of a formal polynomial

p

⁢

(

x

)

about

a

amounts to nothing more than rewriting

p

⁢

(

x

)

in the form

c

0

+

c

1

⁢

(

x

-

a

)

+

⋯

+

c

n

⁢

(

x

-

a

)

n

.

Similar considerations apply to

formal power series

,
or to formal polynomials of several variables.

References

1

Lars V. Ahlfors.

Complex Analysis

, third edition. McGraw-Hill, 1979.

2

Michael Spivak.

Calculus

, third edition. Publish or Perish, 1994.

Title

Taylor series

Canonical name

TaylorSeries

Date of creation

2013-03-22 11:56:59

Last modified on

2013-03-22 11:56:59

Owner

stevecheng (10074)

Last modified by

stevecheng (10074)

Numerical id

24

Author

stevecheng (10074)

Entry type

Definition

Classification

msc 41A58

Classification

msc 26A24

Classification

msc 30B10

Related topic

PowerSeries

Related topic

Analytic

Related topic

TaylorsTheorem

Related topic

ConverseToTaylorsTheorem

Related topic

CalculusOfVariations

Related topic

EulerRelation

Defines

Taylor polynomial

Defines

Taylor expansion

Defines

Maclaurin series

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