numerical sequence and its limit are one of the most important problems in mathematics throughout the history of this science.Is constantly updated knowledge, formulated new theorems and proofs - all this allows us to consider this concept to new positions and from different angles.
numerical sequence, in accordance with one of the most common definition is a mathematical function whose base is the set of natural numbers are arranged according to a particular pattern.
This feature can be considered definite if the law is known, according to which for every natural number can be accurately determine the actual number.
There are several ways to create number sequences.
First, this function can be set so-called "obvious" way, when there is a specific formula by which each member can be determined by simple substitution of numbers in a given sequence.
The second method is called "the recurrent".Its essence lies in the fact that the first few terms are defined numerical sequence, as well as the recurrent special formula by which, knowing the previous member, can be found thereafter.
Finally, the most common way of defining the sequence is the so-called "analytical method" when easily possible to identify not only one or the other member of a certain serial number, but also knowing several successive members come to the general formula given function.
numerical sequence may be increasing or decreasing.In the first case, each followed by its member less than the previous, and the second - on the contrary, more.
Considering this topic, we can not address the question about the limits of sequences.The limit number is called when any, including infinitesimal, there is a sequence number, after which the deviation of consecutive terms of the sequence from a given point in numeric form becomes less than the set value even with the formation of this function.
concept of limit of a numerical sequence is actively used during those or other integral and differential calculation.
mathematical sequences have a whole set of rather interesting properties.
Firstly, any number sequence is an example of a mathematical function, therefore, those properties that are characteristic of the functions can be easily applied to sequences.The most striking example of these properties is the provision of increasing and decreasing the arithmetic series, which are united by one common notion - monotone sequences.
Secondly, there is a fairly large group of sequences that can not be attributed to the increasing nor decreasing - is the periodic sequence.In mathematics, they assumed those functions in which there is the so-called period length, that is, from a certain point (n) begins to act following equation yn = yn + T, where T is and will be the very long period.
