Expansion in powers of x. Solving limits using Taylor series

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Another New Year's Eve... frosty weather and snowflakes on the window glass... All this prompted me to write again about... fractals, and what Wolfram Alpha knows about it. On this occasion there is interesting article, which contains examples of two-dimensional fractal structures. Here we will look at more complex examples of three-dimensional fractals.

A fractal can be visually represented (described) as a geometric figure or body (meaning that both are a set, in this case, a set of points), the details of which have the same shape as the original figure itself. That is, this is a self-similar structure, examining the details of which when magnified, we will see the same shape as without magnification. Whereas in the case of ordinary geometric figure(not a fractal), when zoomed in we will see details that have a simpler shape than the original figure itself. For example, at a high enough magnification, part of an ellipse looks like a straight line segment. This does not happen with fractals: with any increase in them, we will again see the same complex shape, which will be repeated again and again with each increase.

Benoit Mandelbrot, the founder of the science of fractals, wrote in his article Fractals and Art in the Name of Science: “Fractals are geometric shapes, which are equally complex in their details as in their general form. That is, if part of a fractal is enlarged to the size of the whole, it will appear as the whole, either exactly, or perhaps with a slight deformation."

Taylor series. Taylor series expansion of a function.

It turns out that most mathematical functions can be represented with any accuracy in the vicinity of a certain point in the form of power series containing powers of a variable in increasing order. For example, in the vicinity of the point x=1:

By using series called Taylor series, mixed functions containing, say, algebraic, trigonometric, and exponential functions can be expressed as purely algebraic functions. Using series, you can often quickly perform differentiation and integration.

The Taylor series in the neighborhood of point a has the form:

1) , where f(x) is a function that has derivatives of all orders at x=a. R n - the remainder term in the Taylor series is determined by the expression

2)

The k-th coefficient (at x k) of the series is determined by the formula

3) (expansion occurs around the point a = 0)

at a=0

members of the series are determined by the formula

Conditions for using Taylor series.

1. In order for the function f(x) to be expanded into a Taylor series on the interval (-R;R), it is necessary and sufficient that the remainder term in the Taylor (Maclaurin (=McLaren)) formula for this function tends to zero as k →∞ on the specified interval (-R;R).

2. It is necessary that there are derivatives for a given function at the point in the vicinity of which we are going to construct the Taylor series.

Properties of Taylor series.

If f is an analytic function, then its Taylor series at any point a in the domain of definition of f converges to f in some neighborhood of a.

There are infinitely differentiable functions whose Taylor series converges, but at the same time differs from the function in any neighborhood of a. For example:

Taylor series are used in approximation (approximation - scientific method, which consists in replacing some objects with others, in one sense or another close to the original ones, but simpler) functions by polynomials. In particular, linearization ((from linearis - linear), one of the methods for approximate representation of closed nonlinear systems, in which the study does not linear system is replaced by the analysis of a linear system, in a sense equivalent to the original one.) of equations occurs by expanding into a Taylor series and cutting off all terms above the first order.

Thus, almost any function can be represented as a polynomial with a given accuracy.

For students higher mathematics it should be known that the sum of a certain power series belonging to the interval of convergence of the series given to us turns out to be a continuous and unlimited number of times differentiated function. The question arises: is it possible to say that a given arbitrary function f(x) is the sum of a certain power series? That is, under what conditions can the function f(x) be depicted? power series? The importance of this question lies in the fact that it is possible to approximately replace the function f(x) with the sum of the first few terms of a power series, that is, a polynomial. This function replacement is quite simple expression- a polynomial - is also convenient when solving certain problems, namely: when solving integrals, when calculating, etc.

It has been proven that for a certain function f(x), in which it is possible to calculate derivatives up to the (n+1)th order, including the last, in the neighborhood of (α - R; x 0 + R) some point x = α, it is true that formula:

This formula is named after the famous scientist Brooke Taylor. The series that is obtained from the previous one is called the Maclaurin series:

The rule that makes it possible to perform an expansion in a Maclaurin series:

Determine derivatives of the first, second, third... orders.

Calculate what the derivatives at x=0 are equal to.

Write down the Maclaurin series for this function, and then determine the interval of its convergence.

Determine the interval (-R;R), where the remainder of the Maclaurin formula

R n (x) -> 0 at n -> infinity. If one exists, the function f(x) in it must coincide with the sum of the Maclaurin series.

Let us now consider the Maclaurin series for individual functions.

1. So, the first one will be f(x) = e x. Of course, by its characteristics, such a function has derivatives of very different orders, and f (k) (x) = e x , where k equals all. Substitute x = 0. We get f (k) (0) = e 0 =1, k = 1,2... Based on the above, the series e x will look like this:

2. Maclaurin series for the function f(x) = sin x. Let us immediately clarify that the function for all unknowns will have derivatives, in addition, f "(x) = cos x = sin(x+n/2), f "" (x) = -sin x = sin(x +2*n/2)..., f (k) (x) = sin(x+k*n/2), where k equals any natural number. That is, having made simple calculations, we can come to the conclusion that the series for f(x) = sin x will be of the following form:

3. Now let's try to consider the function f(x) = cos x. For all unknowns it has derivatives of arbitrary order, and |f (k) (x)| = |cos(x+k*n/2)|