How to insert mathematical formulas to the website?
If you ever need to add one or two mathematical formulas to a web page, then the easiest way to do this is as described in the article: mathematical formulas are easily inserted onto the site in the form of pictures that are automatically generated by Wolfram Alpha. Besides simplicity, this universal method will help improve site visibility in search engines. It has been working for a long time (and, I think, will work forever), but is already morally outdated.
If you constantly use mathematical formulas on your site, then I recommend that you use MathJax - a special JavaScript library that displays mathematical notation in web browsers using MathML, LaTeX or ASCIIMathML markup.
There are two ways to start using MathJax: (1) using a simple code, you can quickly connect a MathJax script to your site, which will right moment automatically load from a remote server (list of servers); (2) download the MathJax script from a remote server to your server and connect it to all pages of your site. The second method - more complex and time-consuming - will speed up the loading of your site's pages, and if the parent MathJax server becomes temporarily unavailable for some reason, this will not affect your own site in any way. Despite these advantages, I chose the first method as it is simpler, faster and does not require technical skills. Follow my example, and in just 5 minutes you will be able to use all the features of MathJax on your site.
You can connect the MathJax library script from a remote server using two code options taken from the main MathJax website or on the documentation page:
One of these code options needs to be copied and pasted into the code of your web page, preferably between tags and or immediately after the tag. According to the first option, MathJax loads faster and slows down the page less. But the second option automatically monitors and loads the latest versions of MathJax. If you insert the first code, it will need to be updated periodically. If you insert the second code, the pages will load more slowly, but you will not need to constantly monitor MathJax updates.
The easiest way to connect MathJax is in Blogger or WordPress: in the site control panel, add a widget designed to insert third-party JavaScript code, copy the first or second version of the download code presented above into it, and place the widget closer to the beginning of the template (by the way, this is not at all necessary , since the MathJax script is loaded asynchronously). That's all. Now learn the markup syntax of MathML, LaTeX, and ASCIIMathML, and you are ready to insert mathematical formulas into your site's web pages.
Any fractal is constructed according to a certain rule, which is applied sequentially an unlimited number of times. Each such time is called an iteration.
The iterative algorithm for constructing a Menger sponge is quite simple: the original cube with side 1 is divided by planes parallel to its faces into 27 equal cubes. One central cube and 6 cubes adjacent to it along the faces are removed from it. The result is a set consisting of the remaining 20 smaller cubes. Doing the same with each of these cubes, we get a set consisting of 400 smaller cubes. Continuing this process endlessly, we get a Menger sponge.
Expansion of a function into a Taylor, Maclaurin and Laurent series on a site for training practical skills. This series expansion of a function allows mathematicians to estimate the approximate value of the function at some point in its domain of definition. It is much easier to calculate such a function value compared to using the Bredis table, which is so irrelevant in this century computer technology. Expanding a function into a Taylor series means calculating the coefficients before linear functions this series and write it down in correct form. Students confuse these two series, not understanding what is general case, and what is a special case of the second. We remind you once and for all, the Maclaurin series - special case Taylor series, that is, this is the Taylor series, but at the point x = 0. All brief entries for the expansion of well-known functions, such as e^x, Sin(x), Cos(x) and others, are Taylor series expansions , but at point 0 for the argument. For functions of a complex argument, the Laurent series is the most common problem in TFCT, since it represents a two-sided infinite series. It is the sum of two series. We suggest you look at an example of decomposition directly on the website; this is very easy to do by clicking on “Example” with any number, and then the “Solution” button. It is precisely this expansion of a function into a series that is associated with a majorizing series that limits the original function in a certain region along the ordinate axis if the variable belongs to the abscissa region. Vector analysis Another interesting discipline in mathematics is compared. Since each term needs to be examined, the process requires quite a lot of time. Any Taylor series can be associated with a Maclaurin series by replacing x0 with zero, but for a Maclaurin series it is sometimes not obvious to represent the Taylor series in reverse. No matter how much this is required to be done in pure form, but interesting for general self-development. Every Laurent series corresponds to a two-sided infinite power series in integers powers z-a, in other words, a series of the same Taylor type, but slightly different in the calculation of coefficients. We’ll talk about the region of convergence of the Laurent series a little later, after several theoretical calculations. As in the last century, a step-by-step expansion of a function into a series can hardly be achieved simply by reducing the terms to common denominator, since the functions in the denominators are nonlinear. Approximate calculation functional value requires setting tasks. Think about the fact that when the argument of a Taylor series is a linear variable, then the expansion occurs in several steps, but the picture is completely different when the argument of the function being expanded is a complex or nonlinear function, then the process of representing such a function in a power series is obvious, since, in this way Thus, it is easy to calculate, albeit an approximate value, at any point in the definition region, with a minimum error that has little effect on further calculations. This also applies to the Maclaurin series. when you need to evaluate a function in zero point. However, the Laurent series itself is represented here by an expansion on the plane with imaginary units. It will also not be without success correct solution tasks during general process. This approach is not known in mathematics, but it objectively exists. As a result, you can come to the conclusion of the so-called pointwise subsets, and in the expansion of a function in a series you need to use methods known for this process, such as the application of the theory of derivatives. Once again We are convinced that the teacher is right, who made his assumptions about the results of post-computational calculations. Let's note that the Taylor series, obtained according to all the canons of mathematics, exists and is defined on the entire numerical axis, however, dear users of the site service, do not forget the type of the original function, because it may turn out that initially it is necessary to establish the domain of definition of the function, that is, write and exclude from further consideration those points at which the function is not defined in the region real numbers. So to speak, this will show your efficiency in solving the problem. The construction of a Maclaurin series with a zero argument value will not be an exception to what has been said. The process of finding the domain of definition of a function has not been canceled, and you must approach this with all seriousness mathematical operation. In the case of a Laurent series containing the main part, the parameter “a” will be called an isolated singular point, and the Laurent series will be expanded in a ring - this is the intersection of the areas of convergence of its parts, hence the corresponding theorem will follow. But not everything is as complicated as it might seem at first glance to an inexperienced student. Having studied the Taylor series, you can easily understand the Laurent series - a generalized case for expanding the space of numbers. Any series expansion of a function can be performed only at a point in the domain of definition of the function. Properties of functions such as periodicity or infinite differentiability should be taken into account. We also suggest that you use the table of ready-made Taylor series expansions elementary functions, since one function can be represented by up to tens different from each other power series, which can be seen from using our online calculator. Online series Determining Maclaurin is as easy as shelling pears, if you use the site’s unique service, you just need to enter the correct written function and you will receive the presented answer in a matter of seconds, it will be guaranteed to be accurate and in a standard written form. You can copy the result directly into a clean copy for submission to the teacher. It would be correct to first determine the analyticity of the function in question in rings, and then unambiguously state that it is expandable in a Laurent series in all such rings. It is important not to lose sight of the contents negative powers members of the Laurent series. Focus on this as much as possible. Make good use of Laurent's theorem on the expansion of a function in integer powers.
If the function f(x) has derivatives of all orders on a certain interval containing point a, then the Taylor formula can be applied to it:
,
Where r n– the so-called remainder term or remainder of the series, it can be estimated using the Lagrange formula:
, where the number x is between x and a.
f(x)=
At point x 0 =
Number of row elements 3 4 5 6 7
Use the expansion of elementary functions e x , cos(x), sin(x), ln(1+x), (1+x) m
Rules for entering functions:
If for some value X r n→0 at n→∞, then in the limit the Taylor formula becomes convergent for this value Taylor series:
,
Thus, the function f(x) can be expanded into a Taylor series at the point x under consideration if:
1) it has derivatives of all orders;
2) the constructed series converges at this point.
When a = 0 we obtain a series called the Maclaurin series:
,
Expansion of the simplest (elementary) functions in the Maclaurin series:
Exponential functions
, R=∞
Trigonometric functions 
, R=∞ 
, R=∞
, (-π/2< x < π/2), R=π/2
The function actgx does not expand in powers of x, because ctg0=∞
Hyperbolic functions
Logarithmic functions
, -1