The Taylor Series and Taylor Polynomials in Edwards & Penney

presentation of Edwards & Penney's work, by Corry Shores
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We previously examined infinite series with constant terms whose sums (when convergent) were numbers. However, series have more practical usage when the series have variable terms.

We noted before that regarding this equation:

we found:

if we were to write r = x for the ratio in a geometric series, then Theorem 1 gives us as the infinite series representation

of the function f (x) = 1/(1-x). In other words, for each fixed number x with |x| <>1/(1-x). The nth partial sum

of the geometric series in the equation above this one above is now an nth-degree polynomial that approximates the function f (x) = 1/(1 - x). The convergence of the infinite series for

should then be accurate if n is sufficiently large. The figure

Shows the graphs of 1/(1 - x) and the three approximations

It appears that the approximations are more accurate when n is larger and when x is closer to zero.

Polynomial Approximations:

Perhaps we want to calculate or approximate a specific value

of a given function f. We would only need to find a polynomial P (x) whose graph is close to the graph of f on some interval containing

Because if so, we could then use the value

as an approximation of the actual value for

Then as soon as we know how to find such an approximated polynomial P (x), we next would need to know how accurately

approximates the desired value

The simplest example of polynomial approximation is the linear approximation

which we obtained by writing

in the linear approximation formula

The graph for the first-degree polynomial

is the line tangent to the curve y = f (x) at the point (a, f (a)):

This first-degree polynomial agrees with f and with its first derivative at x = a. That is,

Example 2: We suppose that f (x) = ln x and that a = 0. [The ln x means 'the power which e must be raised to get x.'] Then f (1) = 0 [because when x is at point 1, y is at zero] and f ' (1) = 1 [because was determined as the value whose differential is its own value, so when e^x is made just e when x = 1, then we obtain the value for the slope that when multiplied to e obtains e again, which is 1/1]; so,

[because the polynomial takes the a value to be 1]. Hence we expect that

for x near 1 [because we are substituting the polynomial for the function producing e.] With x = 1.1, we find that

Thus the error in this polynomial approximation for ln x is about 5%.

To better approximate ln x near x = 1, we will find a second-degree polynomial

that not only has the same value and the same first derivative as does f at x = 1, but also has the same second derivative there:

In order to satisfy these conditions, we must have:

When we solve these equations we get:

so

With x = 1.1, we see that

which is accurate to three decimal places because

The graph of

is a parabola through (1,0) with the same value, slope, and curvature there as y = ln x:

The tangent line and the parabola used in the computations of this above example illustrate one general approach to polynomial approximation. To approximate the function f (x) near x = a , we look for an nth-degree polynomial

such that its value at a and the values of its first n derivatives at a agree with the corresponding values of f. That is, we require that

We can use these n + 1 conditions to evaluate the values of the n + 1 coefficients

The algebra involved is much simpler, however, if we begin with

expressed as an nth-degree polynomial in powers of x - a rather than in powers of x:

The substituting x = a in the above equation yields

[because the x's cancel to zero, leaving only the first coefficient] by the first condition of the vertical series of equations.

Next, substituting x = a into

yields

so,

[by dividing out the 2 from the left side]. We continue the process to find

In general, the constant term in the kth derivative

because it is the kth derivative of the kth-degree term

(Recall that

denotes the factorial of the positive integer k, read "k factorial"). So when we substitute x = a into

we find that

and thus that

for k = 1, 2, 3, . . . , n.

The above equation also holds for k = 0 if we use the universal convention that 0! = 1 and agree that the zeroth derivative

of the function g is just g itself. With such conventions, our computations establish the following theorem.

Theorem: The nth-Degree Taylor Polynomial:

Suppose that the first n derivatives of the function f (x) exist at x = a. Let

be the nth-degree polynomial

Then the values of

and its first n derivatives agree, at x = a, with the values of f and its first n derivatives there. That is, the equations below hold:

The polynomial in the above equation

is called the nth-degree Taylor polynomial of the function f at the point x = a. We take note that

is a polynomial in powers of x - a rather than in powers of x. To use

effectively for the approximation of f (x) near a, we must be able to compute the value f (a) and the values of its derivatives f ' (a), f '' (a), and so on, all the way to

The line

is simply the line tangent to the curve y = f (x) at the point (a, f (a)). Thus y = f (x) and

have the same slope at this point. We recall from before that the second derivative measures the way the curve y = f (x) is bending as it passes through (a, f (a)). Thus, we call f '' (a) the "concavity" of y = f (x) at (a, f (a)). Then, because

it follows that

has the same value, the same slope, and the same concavity at (a, f (a)) as does y = f (x). In addition,

and f (x) will also have the same rate of change of concavity at (a, f (a)). Such observations suggest that the larger n is, the more closely the nth-degree Taylor polynomial will approximate f (x) for x near a.

Example 3: Find the nth-degree Taylor polynomial of f (x) = 1

The pattern is clear:

hence

so the equation

gives

With n = 2 we obtain the quadratic polynomial:

which is the same as from our previous example. With the third-degree Taylor polynomial

we can go a step further in approximating

The value

is accurate to four decimal places (rounded). In the figure below

we see that the higher the degree and the closer x is to 1, the more accurate the approximation

appears to be.

In the common case a = 0, the nth-degree Taylor polynomial in equation

reduces to

Example 4: Find the nth-degree Taylor polynomial for

This is the easiest of all Taylor polynomials to compute, because

for all

Thus the equation

yields

[by using the nth-Degree Taylor Polynomial theorem:

]. The first few Taylor polynomials of the natural exponential function at a = 0 are, therefore,

The figure below shows the graphs of

The table below shows how these polynomial approximate

for x = 0.1 and for x = 0.5.

At least for these two values of x, the closer x is to a = 0, the more rapidly

appears to approach f (x) as n increases.

Taylor's Formula:

The closeness with which the polynomial

approximates the function f (x) is measure by the difference

for which

This difference

is called the nth-degree remainder for f (x) at x = a. It is the error made if the value f (x) is replaced with the approximation

The theorem that lets us estimate the error, or remainder,

is called Taylor's formula, after Brook Taylor (1685-1731), a follower of Newton who introduced Taylor polynomials in an article published in 1715. The particular expression for

that we give next is called the Lagrange form for the remainder because it first appeared in 1797 in a book by the French mathematician Joseph Louis Lagrange (1936-1813).

Theorem 2: Taylor's Formula:

Suppose that the (n + 1)th derivative of the function f exists on an interval containing the points a and b. Then

for some number z between a and b.

If we replace b with x in the above equation, we get the nth-degree Taylor formula with remainder at x = a:

where z is some number between a and x. Thus the nth-degree remainder term is

[because it is the last term in the series].

Example 3 continued: Estimate the accuracy of the approximation

we substitute x = 1 into the formula

for the kth derivative of f (x) = ln x and get

Hence the third-degree Taylor formula with remainder at x = 1 is

with z between a = 1 and x. With x = 1.1 this gives

where 1 <>z = 1 gives the largest possible magnitude

[more clearly: (0.1)^4/4 = 0.000025] of the remainder term. It follows that

o.095308 <>

so we can conclude that ln(1.1) = 0.0953 to four-place accuracy.

Taylor Series:

If the function f has derivatives of all orders, then we can write Taylor's formula [Suppose that the (n + 1)th derivative of the function f exists on an interval containing the points a and b. Then

for some number z between a and b.] with any degree n that we please. Ordinarily, the exact value of z in the Taylor remainder term in equation

is unknown. Nevertheless, we can sometimes use this equation to show that the remainder approaches zero as

for some particular fixed value of x. Then the equation

gives:

[because

]; that is,

The infinite series:

is called the Taylor Series of the function f at x = a. Its partial sums are the successive Taylor polynomials of f at x = a.

We can write the Taylor series of a function f without knowing that it converges. But if the limit in equation

can be established, then it follows as in equation

that the Taylor Series in equation

actually converges to f (x). If so, then we can approximate the value of f (x) sufficiently accurately by calculating the value of a Taylor polynomial of f of sufficiently high degree.

Example 5: In example 4 we noted that if

for all integers

Hence the Taylor formula

at a = 0 gives

for some z between 0 and x. If x and hence z are negative then

if both are positive. Thus the remainder term

satisfies the inequalities

Therefore, the fact that

for all x implies that

for all x. This means that the Taylor series for

converges to

for all x, and we may write

The series in this above equation is the most famous and most important of all Taylor series. With x = 1, the above equation yields a numerical series

for the number e itself. The 10th and 20th partial sums of this series give the approximations

both of which are accurate to the number of decimal places shown.

from Edwards & Penney: Calculus. New Jersey: Prentice Hall, 2002, p.702a-709c.