1, the definition of mathematical sequence in senior two.
A series of numbers arranged in a certain order is called a series, and each number in the series is called an item of the series.
(1) As can be seen from the definition of series, the numbers of series are arranged in a certain order. If the numbers that make up a series are the same but in different order, then they are not the same series. For example, the series 1, 2,3,4,5 is different from the series 5,4,3,2, 1.
(2) The definition of series does not stipulate that the numbers in the series must be different. Therefore, multiple identical numbers can appear in the same sequence, such as:-1 1 power, 2 power, 3 power, 4 power,? Form a sequence:-1, 1,-1, 1,? .
(4) The term of a series is different from its number. The term of a series refers to a certain number in this series, which is a function value, equivalent to f(n), while the term of a number refers to the position serial number of this number in the series, which is the value of an independent variable, equivalent to n in f(n).
(5) Order is very important for a series. There are several identical figures. Because of their different arrangement order, this series is not the same series. Obviously, there is an essential difference between a series and a group of numbers. For example, if the five numbers 2, 3, 4, 5 and 6 are arranged in different order, you will get different series, and the elements in {2, 3, 4, 5 and 6} are the same set no matter what order they are arranged.
2. The classification of mathematical series in senior two.
(1) According to the number of items in the series, the series can be divided into finite series and infinite series. When writing series, for finite series, the last item should be written, such as series 1, 3, 5, 7, 9,? 2n- 1 means that there is a finite sequence. If the sequence is written as 1, 3, 5, 7, 9,? Or 1, 3, 5, 7, 9, ，2n- 1，？ That represents an infinite sequence.
(2) According to the size relationship between items or the increase or decrease of series, it can be divided into the following categories: increasing series, decreasing series, swinging series and constant series.
3. General formula of mathematical series in senior two.
A sequence is a series of numbers arranged in a certain order, and its essential attribute is to determine the law of this number, which is usually expressed by formula f(n).
Although these two general formulas are different in form, they represent the same series, just as not every functional relationship can be expressed by analytical formula, and not every series can write its general formula. Although some series have general formulas, they are not necessarily unique in form. Only the finite term in front of the series is known, and the series cannot be determined without other explanations, and the general term formula is even more unique. Such as: sequence 1, 2, 3, 4,? ,
The subsequent items written in the formula are different. Therefore, the induction of the general formula should not only look at its first few terms, but also look at the composition law of the series, and observe and analyze more to truly find the internal law of the series. There is no general way to write its general term formula from the first few terms of a series.
In order to understand the general formula of series, emphasize the following points again:
The general formula of (1) series is actually a set of positive integers N* or its finite set {1, 2,? , n} is the expression of the function that defines the domain.
(2) If you know the general formula of the sequence, then use 1, 2, 3,? By substituting n in the formula, we can find the term of this series; At the same time, we can also use the general term formula of series to judge whether a number is an item in the series, and if so, what item it is.
(3) Just as all functional relationships do not necessarily have analytical formulas, not all series have general formulas.
If the approximate value of 2 is insufficient, it shall be accurate to 1, 0. 1,0。 0 1,0。 00 1,0。 000 1,? The order is 1, 1. 4, 1。 4 1, 1。 4 14, 1。 4 14 2,? There is no universal formula.
(4) The general term formulas of some series are not necessarily unique in form, for example:
(5) For some series, only the first few terms are given, but the law of their composition is not given, so the general term formula of series derived from the first few terms is not unique.
4. The image of the mathematical series in Senior Two.
For series 4, 5, 6, 7, 8, 9, 10, the corresponding relationship between the serial number of each item and this item is as follows:
Serial number: 1 2 3 4 5 6 7
Item code: 4 5 6 7 8 9 10
In other words, the above can be regarded as a mapping from one set of serial numbers to another. Therefore, from the perspective of mapping and function, a sequence can be regarded as a set of positive integers N* (or its finite subset {1, 2,3,? , n}), the list of corresponding function values when the independent variables take values in turn from small to large. The function here is a special function, and its independent variable can only take positive integers.
Because the term of the series is the function value and the serial number is the independent variable, the general term formula of the series is the corresponding function and analytical formula.
Sequence is a special function, which can be expressed intuitively by images.
The sequence is represented by images, and a sequence can be represented by a graph with the serial number as the abscissa and the corresponding item as the ordinate. When drawing, for convenience, the unit length taken on the two coordinate axes of the plane rectangular coordinate system can be different, and the change of the sequence can be seen intuitively from the image representation of the sequence, but it is not accurate.
Compared with function, sequence is a special function, which is a group of positive integers or a group of finite continuous positive integers headed by 1, and its image is infinite or finite isolated points.
5. Recursive series of mathematics in senior two.
Finally, I hope that the math knowledge points I compiled last semester will help you, and I wish the students progress in their studies.