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Properties Of Function From Its Maclaurin Series Engineering Essay

The Maclaurin series for any polynomial is the polynomial itself.

The Maclaurin series for (1 − x) −1 is the geometric series

So the Taylor series for x−1 at a = 1 is

By integrating the above Maclaurin series we find the Maclaurin series for −log(1 − x), where log denotes the natural logarithm:

And the corresponding Taylor series for log(x) at a = 1 is

The Taylor series for the exponential function ex at a = 0 is

The above expansion holds because the derivative of ex with respect to x is also ex and e0 equals 1. This leaves the terms (x − 0) n in the numerator and n! In the denominator for each term in the infinite sum.

The Maclaurin series was named after Colin Maclaurin, a professor in Edinburgh, who published the special case of the Taylor result in the 18th century.The function e−1/x² is not analytic at x = 0: the Taylor series is identically 0, although the function is not.

DISCUSSION

If f(x) is given by a convergent power series in an open disc (or interval in the real line) centered at a, it is said to be analytic in this disc. Thus for x in this disc, f is given by a convergent power series

Differentiating term-by-term gives

and so the power series expansion agrees with the Taylor series. Thus a function is analytic in an open disc cantered at a if and only if its Taylor series converges to the value of the function at each point of the disc.

More generally, every sequence of real or complex numbers can appear as coefficients in the Taylor series of an infinitely differentiable function defined on the real line, a consequence of Borel's lemma (see also Non-analytic smooth function Application to Taylor series). As a result, the radius of convergence of a Taylor series can be zero. There are even infinitely differentiable functions defined on the real line whose Taylor series have a radius of convergence 0 everywhere.

Some functions cannot be written as Taylor series because they have a singularity; in these cases, one can often still achieve a series expansion if one allows also negative powers of the variable x; see Laurent series. For example, f(x) = e−x−2 can be written as a Laurent series.

There is, however, a generalization[6]HYPERLINK "http://www.ask.com/wiki/Taylor_series#cite_note-6"[7] of the Taylor series that does converge to the value of the function itself for any bounded continuous function on (0,∞), using the calculus of finite differences. Specifically, one has the following theorem, due to Einar Hill, that for any t > 0,

Here Δnh is the n-th finite difference operator with step size h. The series is precisely the Taylor series, except that divided differences appear in place of differentiation: the series is formally similar to the Newton series. When the function f is analytic at a, the terms in the series converge to the terms of the Taylor series, and in this sense generalizes the usual Taylor series.

In general, for any infinite sequence a i, the following power series identity holds:

So in particular,

The series on the right is the expectation value of f(a + X), where X is a Poisson distributed random variable that takes the value jh with probability e−t/h(t/h)j/j!. Hence,

The law of large numbers implies that that the identity hol

First example

Compute the 7th degree Maclaurin polynomial for the function

.

First, rewrite the function as

.

We have for the natural logarithm (by using the big O notation)

and for the cosine function

The latter series expansion has a zero constant term, which enables us to substitute the second series into the first one and to easily omit terms of higher order than the 7th degree by using the big O notation:

Since the cosine is an even function, the coefficients for all the odd powers x, x3, x5, x7, ... have to be zero.

Second example

Suppose we want the Taylor series at 0 of the function

.

We have for the exponential function

and, as in the first example,

Assume the power series is

Then multiplication with the denominator and substitution of the series of the cosine yields

Collecting the terms up to fourth order yields

Comparing coefficients with the above series of the exponential function yields the desired Taylor series

EXAMPLE-

Find the Maclaurin Series expansion for f(x) = sin x.

ANSWER-

We plot our answer

to see if the polynomial is a good approximation to f(x) = sin x.

We observe that our polynomial (in red) is a good approximation to f(x) = sin x (in blue) near x = 0. In fact, it is quite good between -3 ≤ x ≤ 3.

Finding PI using infinite series-

Leibniz used the series expansion of arctan x to find an approximation of π.

We start with the first derivative:

The value of this derivative when x = 0 is 1.

Similarly for the subsequent derivatives:

f ''(0) = 0

f '''(0) = -2

f iv(0) = 0

f v(0) = 24

Now we can substitute into the Maclaurin Series formula:

Maclaurin series for common functions include

As the degree of the Taylor polynomial rises, it approaches the correct function. This image shows sinx (in black) and Taylor approximations, polynomials of degree 1, 3, 5, 7, 9, 11 and 13.

The exponential function (in blue), and the sum of the first n+1 terms of its Taylor series at 0 (in red).

In mathematics, the Taylor series is a representation of a function as an infinite sum of terms calculated from the values of its derivatives at a single point. The Taylor series was formally introduced by the English mathematician Brook Taylor in 1715. If the series is centered at zero, the series is also called a Maclaurin series, named after the Scottish mathematician Colin Maclaurin who made extensive use of this special case of Taylor's series in the 18th century. It is common practice to use a finite number of terms of the series to approximate a function. The Taylor series may be regarded as the limit of the Taylor polynomials.

The Taylor series of a real or complex function ƒ(x) that is infinitely differentiable in a neighborhood of a real or complex number a is the power series

Which can be written in the more compact sigma notation as?

Where n! Denotes the factorial of n and ƒ (n) (a) denotes the nth derivative of ƒ evaluated at the point a. The zeroth derivative of ƒ is defined to be ƒ itself and(x − a) 0 and 0! Are both defined to be 1? In the case that a = 0, the series is also called a Maclaurin series.

The Greek philosopher Zeno considered the problem of summing an infinite series to achieve a finite result, but rejected it as impossibility: the result was Zeno's paradox. Later, Aristotle proposed a philosophical resolution of the paradox, but the mathematical content was apparently unresolved until taken up by Democritus and then Archimedes. It was through Archimedes's method of exhaustionthat an infinite number of progressive subdivisions could be performed to achieve a finite result. Liu Hui independently employed a similar method a few centuries later.

In the 14th century, the earliest examples of the use of Taylor series and closely-related methods were given by Madhava of Sangamagrama. Though no record of his work survives, writings of laterIndian mathematicians suggest that he found a number of special cases of the Taylor series, including those for the trigonometric functions of sine, cosine, tangent, and arctangent. The Kerala School of astronomy and mathematics further expanded his works with various series expansions and rational approximations until the 16th century.

In the 17th century, James Gregory also worked in this area and published several Maclaurin series. It was not until 1715 however that a general method for constructing these series for all functions for which they exist was finally provided by Brook Taylor, after whom the series are now named.

The Maclaurin series was named after Colin Maclaurin, a professor in Edinburgh, who published the special case of the Taylor result in the 18th century.

The function e−1/x² is not analytic atx = 0: the Taylor series is identically 0, although the function is not.

If f(x) is given by a convergent power series in an open disc (or interval in the real line) centered at a, it is said to be analytic in this disc. Thus for x in this disc, fis given by a convergent power series

Differentiating term-by-term gives

And so the power series expansion agrees with the Taylor series. Thus a function is analytic in an open disc centered at a if and only if its Taylor series converges to the value of the function at each point of the disc.

If f(x) is equal to its Taylor series everywhere it is called entire. The polynomials and the exponential function ex and the trigonometric functions sine and cosine are examples of entire functions. Examples of functions that are not entire include the logarithm, the trigonometric function tangent, and its inverse arctan. For these functions the Taylor series do not converge if x is far from a. Taylor series can be used to calculate the value of an entire function in every point, if the value of the function, and of all of its derivatives, are known at a single point.

Uses of the Taylor series for analytic functions include:

1 The partial sums (the Taylor polynomials) of the series can be used as approximations of the entire function. These approximations are good if sufficiently many terms are included.

2 differentiation and integration of power series can be performed term by term and is hence particularly easy.

An analytic function is uniquely extended to a holomorphic function on an open disk in the complex plane. This makes the machinery of complex analysis available.

The (truncated) series can be used to compute function values numerically, (often by recasting the polynomial into the Chebyshev form and evaluating it with the Clenshaw algorithm).

Algebraic operations can be done readily on the power series representation; for instance the Euler's formula follows from Taylor series expansions for trigonometric and exponential functions. This result is of fundamental importance in such fields as harmonic analysis.

The sine function (blue) is closely approximated by its Taylor polynomial of degree 7 (pink) for a full period centered at the origin.

The Taylor polynomials for log (1+x) only provide accurate approximations in the range −1 < x ≤ 1. Note that, for x > 1, the Taylor polynomials of higher degree are worse approximations.

Pictured on the right is an accurate approximation of sin(x) around the point a = 0. The pink curve is a polynomial of degree seven:

The error in this approximation is no more than |x|9/9! In particular, for −1 < x < 1, the error is less than 0.000003.

In contrast, also shown is a picture of the natural logarithm function log (1 + x) and some of its Taylor polynomials around a = 0. These approximations converge to the function only in the region −1 < x ≤ 1; outside of this region the higher-degree Taylor polynomials are worse approximations for the function. This is similar to Runge's phenomenon.

The error incurred in approximating a function by its nth-degree Taylor polynomial, is called the remainder or residual and is denoted by the function Rn(x).Taylor's theorem can be used to obtain a bound on the size of the remainder.

In general, Taylor series need not be convergent at all. And in fact the set of functions with a convergent Taylor series is a meager set in the Frechet space ofsmooth functions. Even if the Taylor series of a function f does converge, its limit need not in general be equal to the value of the function f(x). For example, the function

Is infinitely differentiable at x = 0, and has all derivatives zero there. Consequently, the Taylor series of f(x) about x = 0 is identically zero. However, f(x) is not equal to the zero function, and so it is not equal to its Taylor series around the origin.

In real analysis, this example shows that there is infinitely differentiable functions f(x) whose Taylor series are not equal to f(x) even if they converge. By contrast in complex analysis there is no complex differentiable functions f (z) whose Taylor series converges to a value different from f (z). The complex function e−z−2does not approach 0 as z approaches 0 along the imaginary axis and its Taylor series is thus not defined there.

More generally, every sequence of real or complex numbers can appear as coefficients in the Taylor series of an infinitely differentiable function defined on the real line. As a result, the radius of convergence of a Taylor series can be zero. There are even infinitely differentiable functions defined on the real line whose Taylor series have a radius of convergence 0 everywhere

Some functions cannot be written as Taylor series because they have a singularity; in these cases, one can often still achieve a series expansion if one allows also negative powers of the variable x; see Laurent series. For example, f(x) = e−x−2 can be written as a Laurent series.

Generalization

There is, however, a generalization of the Taylor series that does converge to the value of the function itself for any bounded continuous function on (0, ∞), using the calculus of finite differences. Specifically, one has the following theorem, due to Einar Hille, that for any t > 0,

Here Δnh is the n-th finite difference operator with step size h. The series is precisely the Taylor series, except that divided differences appear in place of differentiation: the series is formally similar to theNewton series. When the function f is analytic at a, the terms in the series converge to the terms of the Taylor series, and in this sense generalizes the usual Taylor series.

In general, for any infinite sequence ai, the following power series identity holds:

So in particular,

The series on the right is the expectation value of f (a + X), where X is a Poisson distributed random variable that takes the value jh with probability e−t/h (t/h) j/j! Hence,

The law of large numbers implies that that the identity holds.

List of Maclaurin series of some common functions

The real part of the cosinefunction in the complex plane.

An 8th degree approximation of the cosine function in the complex plane.

The two above curves put together.

Several important Maclaurin series expansions follow. All these expansions are valid for complex arguments x.

Exponential function:

Natural logarithm:

Finite geometric series:

Infinite geometric series:

Variants of the infinite geometric series:

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