Numeric sequence: concept, properties, methods of assignment

The numerical sequence and its limit represent one of the most important problems of mathematics throughout the history of the existence of this science. Constantly replenished knowledge, formulated new theorems and proofs - all this allows us to consider this concept from new positions and from different angles.

A numerical sequence, in accordance with one of the most common definitions, is a mathematical function, the basis of which is the set of natural numbers arranged according to one or another regularity.

This function can be considered definite if the law is known, according to which for each natural number it is possible to clearly define a real number.

There are several ways to create numerical sequences.

First, this function can be specified in the so-called "explicit" way, when there is a definite formula by which each of its members can be determined by simply substituting a sequence number into a given sequence.

The second way was called "recurrent". Its essence lies in the fact that the first few terms of the numerical sequence are given, as well as a special recursive formula, with which, knowing the previous term, one can find the next one.

Finally, the most common way of specifying sequences is the so-called "analytical method", when it is easy to not only detect a particular term under a certain sequence number, but also, knowing several consecutive terms, to arrive at a general formula for this function.

A numerical sequence can be decreasing or increasing. In the first case, each subsequent term is less than the previous one, and in the second case, vice versa, more.

Considering this topic, it is impossible not to mention the question about the limits of sequences. The limit of a sequence is a number such that for any, including for an infinitesimal quantity, there is a serial number, after which the deviation of consecutive terms of the sequence from a given point in numerical form becomes less than the value specified even during the formation of this function.

The concept of the limit of a numerical sequence is actively used in the conduct of various integral and differential calculi.

Mathematical sequences have a whole set of quite interesting properties.

First, any numerical sequence is an example of a mathematical function, therefore, those properties that are characteristic of functions can be safely applied to sequences. The most striking example of such properties is the position of increasing and decreasing arithmetic series, which are united by one common notion - monotonous sequences.

Secondly, there is a sufficiently large group of sequences, which can not be attributed to either increasing or decreasing, these are periodic sequences. In mathematics, they are considered to be those functions in which there is a so-called length of the period, that is, from a certain moment (n), the following equality y _n = y _{n + T} begins, where T and will be the same length of the period.