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Taylor’s Theorem
While it is truly elegant that Taylor series exactly converge to functions (on their interval of convergence), it is
the inexact Taylor polynomials that essentially do all of the work.
Let’s ponder this for a minute…
While
sin
x
can be found
exactly
by summing an infinite Taylor series, if we
want to use that information to find
sin 3
, we will have to evaluate Taylor polynomials until we arrive at an
approximation
with which we are satisfied.
Example 1:
Approximating a Function to Specifications
Find a Taylor polynomial that will serve as a substitute for
sin
x
on the interval
,
such that the error
between the Taylor polynomial approximation and
sin
x
is less than 0.0001 for all
x
in
,
.

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P a g e
Goal:
Use Taylor polynomials to approximate functions over the interval of convergence of the
Taylor series while keeping the error of the approximation within specified bounds.
Let’s examine where this error comes from…
Example 2:
Truncation Error for a Geometric Series
Find a formula for the truncation error if we use
2
4
6
1
x
x
x
to approximate
2
1
1
x
over the interval
1,1
.

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P a g e
The Remainder
Every truncation splits a Taylor series into two equally significant pieces:
The Taylor polynomial
n
P
x
that gives us the approximation, and
The
remainder
n
R
x
that tells us whether the approximation is any good
Taylor’s Theorem is about both pieces.
Taylor’s Theorem with Remainder
If
f
has derivatives of all orders in an open interval
I
containing
a
, then for each positive integer
n
and for
each
x
in
I
,
2
...
2!
!
n
n
n
f
a
f
a
f
x
f
a
f
a
x
a
x
a
x
a
R
x
n
where
1
1
1 !
n
n
n
f
c
R
x
x
a
n