Limit the number of columns
Limit the number of columns defined
The concept was born in the limit explore practical problems in precise answers ( such as cutting circle method ).
In an example the cutting circle method, with a circle, a regular hexagon first made therein to its area referred to as \ (A_1 \) ; then make its inscribed regular dodecagon, an area referred to as its z \ (A_2 \) ; then make its inscribed quadrilateral twenty positive, an area referred to as \ (A_3 \) ; such circumstances, each time doubling the number of edges, in general, within the positive contact \ (6 \ times 2 ^ { n -1} \) polygon area referred to as \ (A_n (n-\ in \ mathbb _ + {N}) \) , thus obtaining a series of regular polygon of \ [A_1, A_2, A_3, \ cdots, A_n , \ cdots, \] which constitute a number of orderly when the \ (\ n-) the larger, the smaller the difference between regular polygon and a circle, thereby \ (A_n \) as an approximation of the area of a circle the more accurate but whether \ (n \) take how large, as long as the \ (n \) determines, \ (A_n \) but that is only the polygon area, but not the area of a circle. Therefore, it is envisaged \ (n \) to increase indefinitely (remember as \ (the n-\ to \ infty \) , read as \ (n \) approaches infinity), namely the number of sides of a regular polygon inscribed increase indefinitely, while \ (A_n \)Also infinitely close to a determined value, this value is determined to be understood as the area of the circle. This value is referred to determine which columns have the above order (called mathematically sequence ) \ (A_1, A-2, A- 3, \ cdots, A_n, \ cdots \) when \ (n \ to \ infty \ ) when the limit . in the area of a circle this issue, we see that it is this series of limits was accurately expressed the area of a circle.
This method gradually formed the limit in solving practical problems, has become a basic method of higher mathematics, it is necessary to further clarify.
First described the concept of the number of columns. If If in accordance with a rule for each \ (n-\ in \ mathbb {N} _ + \) , corresponding to a real number determining the \ (x_n \) , the real \ (x_n \) in accordance with the subscript \ (n-\) grew to a sequence of large permutation obtained \ [x_1, x_2, x_3, \ cdots, x_n, \ cdots \] is called the number of columns , abbreviated series \ (\ {x_n \} \ ) .
Each number is called the number of columns in the series is called a term , the \ (n-\) term \ (x_n \) is called the number of columns in general terms (or general term ).
For the problem we have to discuss, the important thing is: when \ (n \) infinitely increases, the corresponding \ (x_n = f (x) \) whether infinitely close to a value determined if we can do? this value is equal to how much?
(The author is the lazy, the direct use of the book examples)
We columns \ [2, \ frac {1 } {2}, \ frac {4} {3}, \ cdots, \ frac {n + (- 1) ^ {n-1}} {n}, \ cdots \ ] analyzed in this series of numbers. \ [x_n = \ FRAC {n-+ (-. 1) ^ {n--. 1}} {n-} =. 1 + \ FRAC {. 1} {n-} (-. 1) ^ {n--. 1 .} \] from the viewpoint of the absolute value, \ (| ab & | \) is smaller, the \ (a \) and \ (B \) closer to \ [| x_n-1 | = | \ frac {1} { n} (- 1) ^ { n-1} |, \] can be found when the \ (\ n-) while growing, \ (\ {n-FRAC {}}. 1 \) smaller, so that \ ( x_n \) is getting closer to \ (1 \) , as long as \ (n-\) is sufficiently large, \ (| x_n-1 | \) may be less than any given positive number, so that when the \ (n-\ ) infinitely increases, \ (x_n \) infinitely close \ (1 \) . for example, a given \ (\ FRAC A} {1} {\) ( \ (A \ in \ mathbb {N} + _ \ )), If you want to \ (|. 1-x_n | <\ FRAC. 1} {A} {\) , as long as the \ (n-> A \) , i.e. from \ (a + 1 \) item starts, can make the inequality . establishment general, whether a given positive number \ (\ varepsilon \) how small, there is always a positive integer \ (N \) , such that when the \ (n N \>) , the inequality \ [| x_n-1 | <\ varepsilon \] have been established which is the number of columns. \ (x_n = \ FRAC {n-+ {-. 1} ^ {n--. 1}} {n-} \) ( \ (n-= 1,2, \ cdots \) ) when \ (n \ to \ infty \ ) infinitely close when \ (1 \) it the substance of such a number. \ (1 \) , called the number of columns \ (x_n = \ frac {n + {- 1} ^ { {}}. 1-n-n-} \) ( \ (n-= 1,2, \ cdots \) ) when \ (x \ to \ infty \ ) when the limit .
In general, there are defined limits the number of breast columns:
Set \ (\ {x_n \} \ ) of a series, if there exists a constant \ (A \) for any given positive number \ (\ varepsilon \) (no matter how small it is) there is always a positive integer \ (N \ ) , such that \ (n> N \) , the inequality \ [| x_n-a | < \ varepsilon \] are true, then it is called constant \ (a \) is the number of columns \ (\ {x_n \} \ ) to limit or the number of columns \ (x_n \) region of convergence \ (A \) , denoted as \ [\ lim_ {n \ to \ infty} x_n = a, \] or \ [x_n \ to a (n \ to \ infty). \]
If this constant is not present \ (A \) , it means \ (\ {x_n \} \ ) no limit, or the number of columns \ (\ {x_n \} \ ) is divergent , and the habits say \ (\ lim_ {n \ to \ infty} \ ) does not exist.
A positive number in the above definition of \ (\ varepsilon \) can be arbitrarily given is important, because there is such inequality \ (| x_n-a | < \ varepsilon \) to express \ (x_n \) and \ (a \) . it should also be infinitely close to the meaning noted: definition positive integer \ (N \) is a positive number with a given \ (\ varepsilon \) relating it with \ (\ varepsilon \) the selection of a given.
On "the number of columns \ (\ {x_n \} \ ) limit is \ (A \) " geometric interpretation here does not expand, are interested can look here .
In order to facilitate the expression, the introduction of the mark " \ (\ FORALL \) " means "for any given" or "for every" sign " \ (\ EXISTS \) " means "presence." Relevant usage and meaning can look here and herein . Sequence limit \ (\ lim_ {n-\ to \ infty} = A \ Leftrightarrow \ FORALL \ varepsilon> 0 \) , \ (\ EXISTS \) positive integer \ (N \) , when the \ (n> N \) when there is \ (| x_n-a | < \ varepsilon \).
Limit the number of columns is defined not directly provide a method of how to find the limit of the number of columns, and later want to find the limit of the law.