The numerical sequence and its limit are one of the most important problems of mathematics throughout the history of this science. Constantly updated knowledge, formulated new theorems and proofs - all this allows us to consider this concept from new perspectives 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 a set of natural numbers, arranged according to one or another pattern.
This function can be considered certain if the law is known, according to which a real number can be clearly determined for each natural number.
There are several options for creating numerical sequences.
Firstly, this function can be defined in the so-called “explicit” way, when there is a certain formula by which each of its members can be determined by simply substituting the sequence number in a given sequence.
The second method is called "recursive". Its essence lies in the fact that the first few members of a numerical sequence are specified, as well as a special recursive formula with which, knowing the previous term, you can find the next one.
Finally, the most general way of defining sequences is the so-called “analytical method”, when without any difficulty it is possible not only to identify a member under a certain serial number, but also, knowing several consecutive terms, to come to a general formula for this function.
The numerical sequence can be decreasing or increasing. In the first case, each subsequent member is smaller than the previous one, and in the second, on the contrary, more.
Considering this topic, one cannot but touch upon the question of the limits of sequences. A sequence limit is a number when, for any, including an infinitesimal quantity, there is a sequence number, after which the deviation of the successive members of the sequence from a given point in a 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 carrying out various integral and differential calculations.
Mathematical sequences have a whole set of rather interesting properties.
Firstly, 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 on increasing and decreasing arithmetic series, which are combined by one common concept - monotone sequences.
Secondly, there is a rather large group of sequences that cannot be attributed to either increasing or decreasing ones - these are periodic sequences. In mathematics, they are considered to be those functions in which there is a so-called period length, that is, from a certain moment (n), the following equality y n = y n + T starts to act, where T will be that period length.