Students of higher mathematics should be aware that the sum of a certain power series belonging to the convergence interval of a given series 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 represented by a power series? The importance of this question is that it is possible to approximately replace f-ju f (x) by the sum of the first few terms of the power series, that is, by a polynomial. Such a replacement of a function by a rather simple expression - a polynomial - is convenient for solving some problems of mathematical analysis, namely, for solving integrals, for calculating differential equations , etc.
It is proved that for some f-ii f (x) in which it is possible to calculate derivatives up to the (n + 1) -th order, including the last, in a neighborhood of (α - R; x 0 + R) of some point x = α, the following formula is valid:
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 decompose in a Maclaurin series:
- Determine the derivatives of the first, second, third ... orders.
- Calculate what the derivatives in x = 0 are equal to.
- Write 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 as n -> infinity. If one exists, the function f (x) in it must coincide with the sum of the Maclaurin series.
We now consider the Maclaurin series for individual functions.
1. So, the first will be f (x) = e x . Of course, in terms of its features, such a function has derivatives of various orders, moreover, f (k) (x) = e x , where k is equal to all natural numbers. Substitute x = 0. We obtain f (k) (0) = e 0 = 1, k = 1,2 ... Based on the foregoing, the series e x will look like this:
2. Maclaurin series for the function f (x) = sin x. We immediately specify that the f-th for all unknowns will have derivatives, moreover, 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 natural number. That is, after making simple calculations, we can conclude that the series for f (x) = sin x will be of this form:
3. Now let's try to consider f-ju f (x) = cos x. For all unknowns, it has derivatives of arbitrary order, and | f
(k) (x) | = | cos (x + k * n / 2) | <= 1, k = 1,2 ... Again, after making certain calculations, we get that the series for f (x) = cos x will look like this:
So, we have listed the most important functions that can be expanded in the Maclaurin series, but they are supplemented by Taylor series for some functions. Now we will list them. It is also worth noting that the Taylor and Maclaurin series are an important part of the series solving workshop in higher mathematics. So, Taylor ranks.
1. The first will be the series for the f-ii f (x) = ln (1 + x). As in the previous examples, for the given f (x) = ln (1 + x), we can add the series using the general form of the Maclaurin series. however, the Maclaurin series can be obtained much more simply for this function. Integrating a certain geometric series, we obtain a series for f (x) = ln (1 + x) of such a sample:
2. And the second, which will be final in our article, will be the series for f (x) = arctg x. For x belonging to the interval [-1; 1], the decomposition is valid:
That's all. In this article, the most used Taylor and Maclaurin series in higher mathematics, in particular, in economic and technical universities, were examined.