The Maclaurin series and the decomposition of certain functions

Students of higher mathematics should be awarethat the sum of a certain power series belonging to the interval of convergence of the series given to us is a continuous and unlimited number of times a differentiated function. The question arises: is it possible to assert that a given arbitrary function f (x) is the sum of a certain power series? That is, under what conditions can the f-ia f (x) be depicted as a power series? The importance of this question lies in the fact that it is possible to approximately replace ffcf (x) with the sum of the first few terms of the power series, that is, the polynomial. Such a replacement of a function by a rather simple expression - a polynomial - is also convenient when solving some problems of mathematical analysis, namely: when solving integrals, calculating differential equations, etc.

It is proved that for some f-II and f (x), in which it is possible to calculate derivatives up to (n + 1) -th order, including the last, in the neighborhood (α - R; x₀ + R) some point x = α is a valid formula:

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

The rule that makes it possible to produce a decomposition in a series of Maclaurin:

1. Determine the derivatives of the first, second, third ... orders.
2. Calculate what the derivatives are equal in x = 0.
3. Write the Maclaurin series for this function, and then determine the interval of its convergence.
4. Determine the interval (-R; R), where the residual part of the Maclaurin formula

R_Mr.(x) -> 0 for n -> infinity. If such exists, in it the function f (x) 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 the most diverse orders, with f^(to)(x) = e^fromwhere k equals all natural numbers. Substitute x = 0. Get f^(to)(0) = e⁰= 1, k = 1,2 ... Based on the above, the row e^x would look like this:

2. The Maclaurin series for the function f (x) = sin x. Immediately clarify 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^(to)(x) = sin (x + k * n / 2), where k equals any natural number. That is, having made simple calculations, we can conclude that the series for f (x) = sin x will be of the following form:

3. Now we will try to consider f-yy f (x) = cos x. It for all unknowns has derivatives of arbitrary order, and | f^(to)(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 features thatcan be decomposed into a Maclaurin series, but they are complemented 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 workshop on solving series in higher mathematics. So, Taylor ranks.

1. The first will be a series for the function f (x) = ln (1 + x).As in the previous examples, for the 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 easier. Integrating a certain geometric series, we obtain a series for f (x) = ln (1 + x) of such a sample:

2. And the second one, which will be final in our article, will be a series for f (x) = arctan x. For x belonging to the interval [-1; 1], the decomposition is valid:

That's all. This article examined the most widely used Taylor and Maclaurin series in higher mathematics, in particular, in economic and technical universities.