Properties Of Function From Its Maclaurin Series Engineering Essay
The Maclaurin series for any polynomial is the polynomial itself.
The Maclaurin series for (1 âˆ’ x) âˆ’1 is the geometric series
So the Taylor series for xâˆ’1 at a = 1 is
By integrating the above Maclaurin series we find the Maclaurin series for âˆ’log(1 âˆ’ 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 aÂ = 0 is
The above expansion holds because the derivative of ex with respect to x is also ex and e0 equalsÂ 1. This leaves the terms (x âˆ’ 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 eâˆ’1/xÂ² is not analytic at xÂ =Â 0: the Taylor series is identically 0, although the function is not.
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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)Â = eâˆ’xâˆ’2 can be written as a Laurent series.
There is, however, a generalizationHYPERLINK "http://www.ask.com/wiki/Taylor_series#cite_note-6" of the Taylor series that does converge to the value of the function itself for any bounded continuous function on (0,âˆž), using the calculus of finite differences. Specifically, one has the following theorem, due to Einar Hill, that for any tÂ >Â 0,
Here Î”nh 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(aÂ +Â X), where X is a Poisson distributed random variable that takes the value jh with probability eâˆ’t/h(t/h)j/j!. Hence,
The law of large numbers implies that that the identity hol
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.
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
Find the Maclaurin Series expansion for f(x) = sin x.
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 â‰¤ x â‰¤ 3.
Finding PI using infinite series-
Leibniz used the series expansion of arctan x to find an approximation of Ï€.
We start with the first derivative:
The value of this derivative when x = 0 is 1.
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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 showsÂ sinxÂ (in black) and Taylor approximations, polynomials of degreeÂ 1,Â 3, 5,Â 7,Â 9,Â 11Â andÂ 13.
TheÂ exponential functionÂ (in blue), and the sum of the firstÂ n+1 terms of its Taylor series at 0 (in red).
InÂ mathematics, theÂ Taylor seriesÂ is a representation of aÂ functionÂ as anÂ infinite sumÂ of terms calculated from the values of itsÂ derivativesÂ at a single point. The Taylor series was formally introduced by the English mathematicianÂ Brook TaylorÂ in 1715. If the series is centered at zero, the series is also called a Maclaurin series, named after the Scottish mathematicianÂ Colin MaclaurinÂ 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 toÂ approximateÂ a function. The Taylor series may be regarded as theÂ limitÂ of theÂ Taylor polynomials.
The Taylor series of aÂ realÂ orÂ complexÂ functionÂ Æ’(x) that isÂ infinitely differentiableÂ in aÂ neighborhoodÂ of aÂ realÂ orÂ complex numberÂ aÂ is theÂ power series
Which can be written in the more compactÂ sigma notationÂ as?
WhereÂ n! Denotes theÂ factorialÂ ofÂ nÂ andÂ Æ’Â (n) (a) denotes theÂ nthÂ derivativeÂ ofÂ Æ’Â evaluated at the pointÂ a. The zeroth derivative ofÂ Æ’Â is defined to beÂ Æ’Â itself and(xÂ âˆ’Â a) 0Â and 0! Are both defined to beÂ 1? In the case thatÂ aÂ = 0, the series is also called aÂ Maclaurin series.
The Greek philosopherÂ ZenoÂ considered the problem of summing an infinite series to achieve a finite result, but rejected it as impossibility: the result wasÂ Zeno's paradox. Later,Â AristotleÂ proposed a philosophical resolution of the paradox, but the mathematical content was apparently unresolved until taken up byÂ DemocritusÂ and thenÂ Archimedes. It was through Archimedes'sÂ method of exhaustionthat an infinite number of progressive subdivisions could be performed to achieve a finite result.Â Liu HuiÂ 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 byÂ Madhava of Sangamagrama.Â Though no record of his work survives, writings of laterIndian mathematiciansÂ suggest that he found a number of special cases of the Taylor series, including those for theÂ trigonometric functionsÂ of sine, cosine,Â tangent, andÂ arctangent. TheÂ Kerala School of astronomy and mathematicsÂ further expanded his works with various series expansions and rational approximations until the 16th century.
In the 17th century,Â James GregoryÂ 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 byÂ Brook Taylor,Â after whom the series are now named.
The Maclaurin series was named afterÂ Colin Maclaurin, a professor in Edinburgh, who published the special case of the Taylor result in the 18th century.
The functionÂ eâˆ’1/xÂ²Â is not analytic atxÂ =Â 0: the Taylor series is identically 0, although the function is not.
IfÂ f(x) is given by a convergent power series in an open disc (or interval in the real line) centered atÂ a, it is said to beÂ analyticÂ in this disc. Thus forÂ xÂ in this disc,Â 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 atÂ aÂ if and only if its Taylor series converges to the value of the function at each point of the disc.
IfÂ f(x) is equal to its Taylor series everywhere it is calledÂ entire. The polynomials and theÂ exponential functionÂ exÂ and theÂ trigonometric functionsÂ sine and cosine are examples of entire functions. Examples of functions that are not entire include theÂ logarithm, theÂ trigonometric functionÂ tangent, and its inverseÂ arctan. For these functions the Taylor series do notÂ convergeÂ ifÂ xÂ is far fromÂ 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.
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Uses of the Taylor series for analytic functions include:
1 The partial sums (theÂ 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.
AnÂ analytic functionÂ is uniquely extended to aÂ holomorphic functionÂ on anÂ open diskÂ in theÂ complex plane. This makes the machinery ofÂ complex analysisÂ available.
The (truncated) series can be used to compute function values numerically, (often by recasting the polynomial into theÂ Chebyshev formÂ and evaluating it with theÂ Clenshaw algorithm).
Algebraic operations can be done readily on the power series representation; for instance theÂ Euler's formulaÂ follows from Taylor series expansions for trigonometric and exponential functions. This result is of fundamental importance in such fields asÂ 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 rangeÂ âˆ’1 <Â xÂ â‰¤ 1. Note that, forÂ xÂ > 1, the Taylor polynomials of higher degree areÂ worseÂ approximations.
Pictured on the right is an accurate approximation of sin(x) around the pointÂ aÂ = 0. The pink curve is a polynomial of degree seven:
The error in this approximation is no more than |x|9/9! In particular, forÂ âˆ’1 <Â xÂ < 1, the error is less than 0.000003.
In contrast, also shown is a picture of the natural logarithm functionÂ log (1 +Â x)Â and some of itsÂ Taylor polynomialsÂ aroundÂ aÂ = 0. These approximations converge to the function only in the region âˆ’1 <Â xÂ â‰¤ 1; outside of this region the higher-degree Taylor polynomials areÂ worseÂ approximations for the function. This is similar toÂ Runge's phenomenon.
TheÂ errorÂ incurred in approximating a function by itsÂ nth-degree Taylor polynomial, is called theÂ remainderÂ orÂ residualÂ and is denoted by the functionÂ Rn(x).Taylor's theoremÂ can be used to obtain a bound on the size of the remainder.
In general, Taylor series need not beÂ convergentÂ at all. And in fact the set of functions with a convergent Taylor series is aÂ meager setÂ in theÂ Frechet spaceÂ ofsmooth functions. Even if the Taylor series of a functionÂ fÂ does converge, its limit need not in general be equal to the value of the functionÂ f(x). For example, the function
IsÂ infinitely differentiableÂ atÂ xÂ = 0, and has all derivatives zero there. Consequently, the Taylor series ofÂ f(x) aboutÂ xÂ = 0Â is identically zero. However,Â f(x) is not equal to the zero function, and so it is not equal to its Taylor series around the origin.
InÂ real analysis, this example shows that there isÂ infinitely differentiable functionsÂ f(x) whose Taylor series areÂ notÂ equal toÂ f(x) even if they converge. By contrast inÂ complex analysisÂ there isÂ noÂ complex differentiable functionsÂ f (z) whose Taylor series converges to a value different fromÂ f (z). The complex function eâˆ’zâˆ’2does not approach 0 asÂ zÂ 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, theÂ radius of convergenceÂ 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 aÂ singularity; in these cases, one can often still achieve a series expansion if one allows also negative powers of the variableÂ x; seeÂ Laurent series. For example,Â f(x)Â = eâˆ’xâˆ’2Â can be written as a Laurent series.
There is, however, a generalizationÂ of the Taylor series that does converge to the value of the function itself for anyÂ boundedÂ continuous functionÂ on (0, âˆž), using the calculus ofÂ finite differences. Specifically, one has the following theorem, due toÂ Einar Hille, that for anyÂ tÂ >Â 0,
Here Î”nhÂ is theÂ n-th finite difference operator with step sizeÂ 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 functionÂ fÂ is analytic atÂ 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 sequenceÂ ai, the following power series identity holds:
So in particular,
The series on the right is theÂ expectation valueÂ ofÂ f (aÂ +Â X), whereÂ XÂ is aÂ Poisson distributedÂ random variableÂ that takes the valueÂ jhÂ with probabilityÂ eâˆ’t/h (t/h) j/j! Hence,
TheÂ law of large numbersÂ implies that that the identity holds.
List of Maclaurin series of some common functions
The real part of the cosinefunction in theÂ complex plane.
An 8th degree approximation of the cosine function in theÂ complex plane.
The two above curves put together.
Several important Maclaurin series expansions follow.Â All these expansions are valid for complex argumentsÂ x.
FiniteÂ geometric series:
Infinite geometric series:
Variants of the infinite geometric series:
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