Category: Sequence and Limit

Number Sequence

In physics and other sciences there are many quantities: time, length, volume etc. Some of them are variable (take different values) and some are constant (take constant values).

However, in mathematics we abstract from physical sense of quantity, we are interested only in numbers that this quantity can take. Physical sense becomes important in applications of mathematics. Therefore, for us variable quantity (or simple variable) is number variable. It is denotes by some symbol (for example, $x$) to which we attach numeric value.

Limit of a Sequence

Often we are interested in value that sequence will take as number ${n}$ becomes very large.

Definition. Constant number ${a}$ is called a limit of the sequence ${x}_{{n}}$ if for every $\epsilon>{0}$ there exists number ${N}$, such that all values ${x}_{{n}}$ whose number ${n}>{N}$, satisfy inequality ${\left|{x}_{{n}}-{a}\right|}<\epsilon$.

Infinitely Small Sequence

Definition. Sequence ${x}_{{n}}$ is called infinitesimal if its limit is 0.

In other words, if for every $\epsilon>{0}$ there is number ${N}={N}_{{\epsilon}}$, such that for ${n}>{N}$, ${\left|{x}_{{n}}-{0}\right|}={\left|{x}_{{n}}\right|}<\epsilon.$

Infinitely Large Sequence

Definition. Sequence ${x}_{{n}}$ is called infinitude if for every ${E}>{0}$ we can find such number ${N}_{{E}}$ that ${\left|{x}_{{n}}\right|}>{E}$.

We can reformulate definition as follows: sequence ${x}_{{n}}$ is infinitude if its absolute value becomes more than some specified number ${E}>{0}$, starting with some number. In other words infinitude grows without a bound when n becomes large. For example, for sequence ${x}_{{n}}={{n}}^{{2}}$ ${x}_{{1000}}={1000000}$ and it will take even larger values when ${n}$ becomes larger.