The Maclaurin series and the decomposition of certain functions

The student of higher mathematics should know that the sum of some power series belonging to the interval of convergence of the given series is a differentiated function that is continuous and infinitely many times. The question arises: is it possible to assert that a given arbitrary function f (x) is the sum of a power series? That is, under what conditions can the f-th f (x) be represented by a power series? The importance of such a question is that it is possible to approximately replace f-x f (x) by the sum of several first terms of the power series, that is, a polynomial. Such a substitution of a function by a rather simple expression-a polynomial-is also convenient in solving certain problems of mathematical analysis, namely: in solving integrals, in calculating differential equations, and so on.

It is proved that for some function f (x), in which it is possible to calculate the derivatives up to (n + 1) -th order, including the last one, in a neighborhood of (α - R; X ₀ + R) of some point x = α, the following formula is valid:

This formula bears the name of the famous scientist Brooke Taylor. The series that is obtained from the previous one is called the Maclaurin series:

A rule that makes it possible to decompose into a Maclaurin series:

1. Determine the derivatives of the first, second, third ... orders.
2. Calculate what the derivatives at x = 0 are equal to.
3. Record the Maclaurin series for a given function, and then determine the interval of its convergence.
4. Determine the interval (-R; R), where the remainder of the Maclaurin formula

R _n (x) -> 0 as n → ∞ of infinity. In the case where it 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. Thus, the first is f (x) = e ^x . Of course, in terms of its characteristics, such a function has derivatives of very different orders, and f ^(k) (x) = e ^x , where k equals all natural numbers. We substitute x = 0. We obtain f ^(k) (0) = e ⁰ = 1, k = 1,2 ... Proceeding from the foregoing, the series e ^x Will look like this:

2. The Maclaurin series for the function f (x) = sin x. We immediately clarify that the φ-function for all unknowns has 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, by making simple calculations, we can come to the conclusion that the series for f (x) = sin x will be of the form:

3. Now let's try to consider the function f (x) = cos x. It has derivatives of arbitrary order for all unknowns, and | f ^(k) (x) | = Cos (x + k * n / 2) | <= 1, k = 1,2 ... Again, making certain calculations, we get that the series for f (x) = cos x will look like this:

So, we listed the most important functions that can be decomposed into the Maclaurin series, but they are complemented by Taylor series for some functions. Now we list them. It is also worth noting that the Taylor and Maclaurin series are an important part of the workshop of solving series in higher mathematics. So, the Taylor series.

1. The first is the series for the function f (x) = ln (1 + x). As in the previous examples, for a given f (x) = ln (1 + x) we can add a series using the general form of the Maclaurin series. However, for this function the Maclaurin series can be obtained much simpler. Integrating some 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 paper, will be a series for f (x) = arctg x. For x belonging to the interval [-1; 1], the expansion is valid:

That's all. In this article, the most used series of Taylor and Maclaurin in higher mathematics, in particular, in economic and technical universities, were considered.