Taylor polynomials.
Definition 10.6.1
Let

be a function differentiable at least

times in a neighborhood of

. The polynomial
is called
the Taylor polynomial of degree

of the function

at

.
If
, this polynomial is called the Mac-Laurin polynomial of degree
of
.
Example 10.6.2
Let

.
For any natural number
, we have
, thus
.
Thus the MacLaurin series of
is:
The Taylor polynomial of order

at 0 of

is therefore:
The number
is called Lagrange's remainder.
This theorem gives an estimation of the error when approximating a function by its Taylor polynomial of order
.
Theorem 10.6.4 (The Remainder Estimation Theorem)
- With the settings of Thm 6.3, suppose that there exist two positive numbers
and
such that for any
between
and
, the following inequality is verified:
Then, Lagrange's remainder verifies the inequality:
- If all these conditions hold for every natural number
, then Taylor series converges and its sum is equal to
.
Example 10.6.5
Let

. We have:
Therefore

and

.
We have:
As all values of the cosine function are between

and 1, the remainder

verifies the inequality:
As

, for all real

, the MacLaurin series of
the cosine function is convergent for every real number

.
In Fig. 2 is a Maple display for MacLaurin polynomials of the cosine function (blue), of order 2 (yellow curve),4 (green curve) and 6 (red curve). Pay attention to the fact it seems that the larger the degree of the approximation, the better the approximation.
Figure 2:
The graphs of the first MacLaurin polynomials for the cosine function.
 |
As
, for all real
, the MacLaurin series of
the cosine function is convergent for every real number
.
By the same way we can prove that all the MacLaurin series in 5.2 converge towards the
given functions.
In Fig. 3 is a Maple display for MacLaurin polynomials of the sine function (in blue), of order 1 (in green),3 (in red) and 5 (in yellow).
Figure 3:
The graphs of the first MacLaurin polynomials for the sine function.
 |
Finally, we give in Fig. 4 a mathplot display for MacLaurin polynomials of the function
such
that
of order 1 (curve
),2 (curve
) and 3 (curve
).
Figure 4:
The graphs of the first MacLaurin polynomials for
.
 |
Studying the convergence domain of a Taylor series is a non trivial issue. Let us see a couple of examples.
Take
. The function
is defined over
. Its first MacLaurin polynomials are given as follows:
The MacLaurin series of the function
is a geometric series whose ratio is equal to
, thus it is convergent iff
. It has no meaning (at least for us now) out of this interval. This situation is illustrated in Figure 5.
Figure 5:
The MacLaurin series converges on a subdomain of the function's domain.
 |
We can find a first partial explanation for this situation: at -1, the function
is not defined. As the domain of convergence of the MacLaurin series is an interval centered at 0, it cannot be larger than
.
Take now
. This function is defined over the whole of
.The first partial sums of its MacLaurin series are as follows:
The series is geometric and its ratio is equal to
. Thus as previously the series is convergent iff
. The graphs of the function and of these partial sums are displayed in Figure 6.
Figure 6:
The MacLaurin series converges on a subdomain of the function's domain.
 |
The function
is defined over the whole of
. There is no opportunity to use an argument as the one used above for
. We will wait until we learn the course in Complex Variables to have a more convincing argument (http://ndp.jct.ac.il/tutorials/complex/node51.html).
Noah Dana-Picard
2007-12-28