The number sequence and its limit have been one of the most important problems in 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 positions and from different angles.
A number 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 pattern or another.
This function can be considered defined if the law is known, according to which a real number can be clearly defined for each natural number.
There are several options for creating number 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 determinedby simple substitution of the serial number in the given sequence.
The second method is called "recurrent". Its essence lies in the fact that the first few members of the numerical sequence are given, as well as a special recursive formula, with the help of which, knowing the previous member, you can find the next one.
Finally, the most general way of specifying sequences is the so-called "analytical method", when without much difficulty one can not only identify one or another term under a certain serial number, but also, knowing several consecutive terms, come to the general formula of a given functions.
The number sequence can be decreasing or increasing. In the first case, each subsequent term is less than the previous one, and in the second case, on the contrary, it is greater.
Considering this topic, it is impossible not to touch upon the issue of the limits of sequences. The limit of a sequence is such a number when for any value, including an infinitesimal one, there is a serial number after which the deviation of successive members of the sequence from a given point in numerical form becomes less than the value specified during the formation of this function.
The concept of the limit of a numerical sequence is actively used when carrying out certain integral and differential calculations.
Mathematical sequences have a whole set of quite interestingproperties.
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 provision on increasing and decreasing arithmetic series, which are united by one common concept - monotonic sequences.
Secondly, there is a fairly large group of sequences that cannot be classified as either increasing or decreasing - 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 begins to operate y =yn+T, where T will be the very length of the period.
