- Return to the Power Series starting page
- Representing functions as power series
- A list of common Maclaurin series
- Error Bounds using Taylor Polynomials

A little examination using derivatives brings the following conclusion:

If *f* has a power series representation at *a*, that is,

then

.

In other words,

.

This series is called the Taylor series of *f* at
*a*. In the
special case
where *a*=0, this is called the Maclaurin series. Of course,
the statement
"if *f* has a power series representation" is an important one. For
some functions,
you can create the above series, but it will not converge to the function
value. Functions with a power series representation at a point *a*
are called
analytic at *a*.

In order for *f* to be analytic, we want *f* to equal its power
series, or,
in other words, to be the limit of the sequence of partial sums. The partial
sums are

.

The term *T*[*n*](*x*) is called the
*n*th-degree
Taylor polynomial of *f* at *a*.
We want *f*(*x*)=lim *T*[*n*](*x*), or we want
*R*[*n*](*x*) to go to zero as *n* gets large,
where
*R*[*n*](*x*)=*f*(*x*)-*T*[*n*](*x*).

For instance, since we know that

,

the 5th degree Taylor polynomial of 1/(1-x) is

.

To help us determine analyticity(?), we have the following theorem:

Iff(x)=T[n](x)+R[n](x), whereT[n](x) is thenth-degree Taylor polynomial offata, and the limit ofR[n](x) asngoes to infinity is 0 for |x-a|<R, thenfis equal to the sum of its Taylor series on the interval |x-a|<R, that is,fis analytic ata.

A helpful result for all this is the following:

Iffhasn+1 derivatives in an intevalIthat contains the numbera, then forxinIthere is a numberzstrictly betweenxandasuch that the remainder term in the Taylor series can be expressed as

.

This makes finding the limit of *R*[*n*](*x*) much easier.
Remember, that *z* in this formula depends on *x*; namely, it
must be between *a* and *x*.

- Return to the Power Series starting page
- Representing functions as power series
- A list of common Maclaurin series
- Error Bounds using Taylor Polynomials

Copyright © 1996 Department of Mathematics, Oregon State University

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