Before getting to the point what the taylor and maclaurin series are, you need to clearly understand what the number series are, as well as you need information about power series and its convergence area.
Basic Definitions before Getting to the Maclaurin Series
The solution of a problem presented in mathematical terms, for example, in a combination of various functions, their derivatives and integrals, should lead to the number, which is most often the final answer. In order to do this, there are different methods developed in different areas of mathematics.
The theory of series, including maclaurin series is the section of mathematics, which allows solving any correctly put problem with sufficient accuracy for practical use.
Even if some of the finer concepts of mathematical analysis have appeared out of touch with the theory of series, they were immediately applied to the series, which served as a tool to test the relevance of these concepts. This situation persists today as well.
If the members of the series are:
- The numbers, the series is called numerical.
- The numbers of the same sign, then the series is called the constant sign series.
- The numbers of different signs, then the series is called alternating series.
- Positive numbers, then the series is called positive series numbers.
- The numbers, which signs are strictly alternating, then the series is called an alternating.
- The functions, then the series is called functional.
- The degrees, then the series is called power series.
- The trigonometric functions, then the series is called trigonometric.
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The Definition of the Power Series
The functional series of the form a0, a1, a2, … do not depend on the variable z and called power series that don’t depend on the z variable. The numbers a0, a1, a2, … are the coefficients of this series.
As it is usually clear, which variable defines the functional series as the power series, we will continue to speak simply about power series.
As in the case of general functional series, we can talk about real and complex power series.
If the variable z can take the complex (and real) values, and the coefficients of the series are complex numbers, then the power series is called complex.
If the values of the variable z can only be real, and the coefficients of the series are also real numbers, then the power series is called real.
The intermediate case when the value of the variable z must be real and coefficients of the series may be complex (an = bn + icn) are not of great interest. Generally, in this case all the information about the following series: (b0 + ic0) + (b1 +ic1) z +… + (bn + icn) zn +… can be obtained by considering separately two real series: b0 + b1z + … + bnzn +… and c0 + c1z + … + cnzn + …
Real Power Series and of Convergence Interval
If the series a0 + a1x + … + anxn + … has real coefficients and variable x takes only real values, then the theorem of Abel leads to the following statement:
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The Steady Convergence of the Series in the Circle of Its Convergence
Theorem: the power series converges uniformly in any closed circle contained in its circle of convergence. This thesis format can be proven by the following: let’s say the a0 + a1z + … + anzn + … is the power series and R is the radius of convergence. Take an arbitrary vicious circle that lies inside the circle of convergence. Obviously, we can assume that the smaller circle center is also lies at 0. More precisely, every smaller circle can be taken by the circle with the center at 0 and that is entirely contained in the circle of convergence. The uniform convergence of the series in the taken circle leads to the uniform convergence in a smaller circle as well.
Expansion of Functions in Power Series
According to the case study template, the sum of any convergent power series is a function defined inside the circle of the convergence of the series (and perhaps even in some points of its boundary).
In this regard, there are two problems. Firstly, with the given series it is possible to search for the function that is equal to the series amount in the place of its convergence. This problem is called the summation of convergent series. Secondly, with the given series it is possible to search the convergent series of a particular type, the amount of which in the area of the convergence would be equal to a given function. This problem is called the expansion of functions in a series.
Along with power series in regard to the variable z, i.e. types of series as a0 + a1z + … + anzn + … it will be convenient to consider the power series in the regard to the variable z–a like a0 + a1 (z–a) + … + an (z–a)n + …
Clearly, the substitution of the y = z–a, the second of the series is converted into the first series. Therefore, if the circle of convergence of the first series consists of all points for which |z|<= R, then for the same reasons, the circle of convergence of the second series consists of all y points, for which |y|<=R i.e., |z–a| <=R. In other words, in the complex plane, where the independent variable z is represented, the circle of convergence of the series has the same radius R as the other circle of convergence of the series, and its center is located at the point a.
Specifically, if the both series are real, then the interval of convergence of one series is obtained by shifting the interval of the convergence of the other series for a points to the right (obviously if the a<0 it is the shift to the left).
The Taylor and Maclaurin Series
If the function f (x) has in a certain segment the derivatives of all orders (since they are all present, each of them will be differentiable and therefore continuous), then in the science paper we can write the Taylor formula for any value of n.
The Taylor series is the expansion of the function in the endless amount of power functions.
Taylor series were named after the British mathematician Taylor, although the Taylor series was known long before Taylor’s publications. It was used back in the XVII century by Gregory and Newton.
Terms of the Taylor Series Application:
- In order for the function f(x) to be expanded in a Taylor series in the interval (–R;R) it is necessary and sufficient that the residual member in the Taylor formula for this function tends to zero as k→∞ on the mentions interval (–R;R).
- It is necessary for the derivatives to be present for the function at a point, in the vicinity of which we are going to build a Taylor series.
From a mathematical point of view, the infinite length of such a series is not an obstacle for its consideration and further analytical transformations. From the viewpoint of actual calculations, the series has to be limited to a finite number of lengths. In other words, it is necessary to limit the upper index of Taylor series with some finite value of N.
The Taylor Series Qualities:
- If f is an analytic function, then its Taylor series (maclaurin series ) at any a point of the f area converges to f in the vicinity of a.
- If there are infinitely differentiable functions, then Taylor series (maclaurin series ), which converges, but at the same time differs from the function in any vicinity of a.
Taylor series are used for function approximation with polynomials (approximation is the scientific method, which implies the replacement of some objects with other objects, in one sense or another the objects are close to the source ones, but they are much simpler). In particular, Taylor series are applied for the linearization (linear), a method of approximate representation of the closed nonlinear systems, in which the study of non-linear system is replaced by the analysis of the linear system, in a sense of equivalent to the original. Equations occur by the expansion in Taylor series and cut off all the members of the above first order.
Thus, almost any function can be represented as a polynomial with a given accuracy.
The Definition of Maclaurin Series
Maclaurin series were named after the leading English mathematician Colin Maclaurin, a professor at Edinburgh University, a follower of Newton, with whom he was personally acquainted. In his famous work Maclaurin talked about maclaurin series. However, the maclaurin series was not a new discovery, as it was mentioned by Brook Taylor. But even earlier the maclaurin series was discovered by Johann Bernoulli and, in fact, Leibniz also knew about it. Maclaurin acknowledged that he was fully obligated to Taylor for the maclaurin series.
Maclaurin series is a Taylor series in the vicinity of a point a=0. The classic derivation of coefficients of a power series in the form of the Maclaurin series is produced by analytical differentiation of a power series recorded in a general way.
It turns out that the majority of practically encountered mathematical functions can be represented with any accuracy in the vicinity of a point in the form of power series (maclaurin series ), containing degrees of a variable in an ascending order. For example, in the vicinity of a point x=0.
When using the so called maclaurin series, the mixed function, containing, say, algebraic, trigonometric, and exponential functions can be expressed as a purely algebraic functions. With the maclaurin series it is possible to quickly implement differentiation and integration.
Maclaurin series are a special case of the Taylor series.
The Terms of the Maclaurin Series Use:
- In order for the function f(x) to be expanded in a Maclaurin series in the interval (–R; R) it is necessary and sufficient that the residual members in the Maclaurin formula for this function tends to zero if k →∞ on the specified range (–R; R).
- It is necessary for the derivatives to be present for the function at the point a=0, in the vicinity of which we are going to build a Maclaurin series.
Numerical Integration Using a Maclaurin Series
The values of many integrals can’t be found using any analytical methods. You should already know about the calculation of such integrals using the trapezoidal formula and Simpson formula. Another method of finding the numerical value of a definite integral is the expression of the function in the form of the Maclaurin series followed by alternately integration of each member.
As a rule, many of the functions are approximated by Maclaurin series that in some way simplifies the calculation of the derivatives of these functions and enables to get more compact forms of recording series. A striking example is the approximation of the exponential function by the Maclaurin series.
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