studied advanced mathematics should be known that the sum of a power series in the interval of convergence of a number of us, is a continuous and unlimited number of times differentiated function.The question arises: is it possible to argue that given an arbitrary function f (x) - is the sum of a power series?That is, under what conditions the f-Ia f (x) can be represented by a power series?The importance of this issue is that it is possible to replace approximately Q-uw f (x) is the sum of the first few terms of a power series, that is polynomial.Such a replacement function is quite simple expression - polynomial - is convenient and in solving certain problems in mathematical analysis, namely in solving integrals in calculating differential equations, and so on. D.
proved that for some f-ii f (x)which can calculate the derivatives of the (n + 1) th order, including the latest, in the vicinity of (α - R; x0 + R) of a point x = α is a fair formula:
This formula is named after the famous scientist Brooke Taylor.The series, which is derived from the previous one, called a Maclaurin series:
rule that makes it possible to produce a Maclaurin series expansion:
- Determine the derivatives of the first, second, third ... order.
- calculated, which are derivatives in x = 0.
- Record Maclaurin series for this function, and then to determine the interval of convergence.
- determine the interval (-R; R), where the remainder of the Maclaurin formula
Rn (x) - & gt;0 for n - & gt;infinity.If it exists, it function f (x) must be equal to the sum of the Maclaurin series.
Consider now the Maclaurin series for the individual functions.
1. Thus, the first is f (x) = ex.Of course, by their characteristics such f-Ia has derivatives of a variety of orders, and f (k) (x) = ex, where k is equal to all the natural numbers.Substituting x = 0.We get f (k) (0) = e0 = 1, k = 1,2 ... Based on the above, a number of ex will be as follows:
2. Maclaurin series for the function f (x) = sin x.Immediately specify that f-Ia for all unknowns will have derivatives besides 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 is equal to any positive integer.That is, by performing simple calculations, we can conclude that the series for f (x) = sin x is of this type:
3. Now let's consider the Theological Faculty of f (x) = cos x.It is for all of the unknown has derivatives of arbitrary order, and | f (k) (x) | = | cos (x + k * n / 2) | & lt; = 1, k = 1,2 ... yet again, producingcertain calculations, we find that the series for f (x) = cos x would look like this:
So, we have listed the most important features that can be expanded in a Maclaurin series, but they complement the Taylor series for some functions.Now we will list them as well.It should also be noted that Taylor and Maclaurin series are an important part of the workshop series in solutions of higher mathematics.So, Taylor series.
1. The first is the series for f-ii f (x) = ln (1 + x).As in the previous examples, for this we f (x) = ln (1 + x) can be folded in a row, using the general form of Maclaurin series.However, this function Maclaurin can be obtained much easier.Integrating a geometric series, we get the series for f (x) = ln (1 + i) of the sample:
2. And the second, which will be final in this article, is the series for f (x) = arctg's.For x belonging to the interval [-1, 1] is the expansion of the fair:
That's all.In this article we were considered the most used Maclaurin and Taylor series in higher mathematics, in particular in the economic and technical colleges.
