Infinitely Large Sequence

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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.

If sequence is infinitude and for at least large values of ${n}$ preserves sign (+ or -), then according to the sign we say that sequence ${x}_{{n}}$ has limit $+\infty$ or $-\infty$ and write: $\lim{x}_{{n}}=+\infty,\lim_{{{n}\to+\infty}}{x}_{{n}}=+\infty,{x}_{{n}}\to+\infty$ or $\lim{x}_{{n}}=-\infty,\lim_{{{n}\to+\infty}}{x}_{{n}}=-\infty,{x}_{{n}}\to-\infty$. Also we say that sequence has infinite limit.

We already wrote that numbers $\pm\infty$ represent very large and very small numbers. But they are not numbers in a full sense of this word. They just a way to write very large (small) numbers shortly. Arithmetic operations on these numbers are not performed, because we don't know what is $\infty-\infty$.

Also, $+\infty$ is often written as $\infty$.

Example 1. Consider sequences ${x}_{{n}}={n}$, ${x}_{{n}}=-{n}$, ${x}_{{n}}={{\left(-{1}\right)}}^{{{n}+{1}}}{n}$.

Corresponding lists are

${\left\{{1},{2},{3},{4},\ldots\right\}}$,

${\left\{-{1},-{2},-{3},-{4},\ldots\right\}}$,

${\left\{{1},-{2},{3},-{4},\ldots\right\}}$.

All variants are infinitude because ${\left|{x}_{{n}}\right|}={\left|{n}\right|}>{E}$ when ${n}>{E}$. Therefore, we can take ${N}_{{E}}>{\left[{E}\right]}$, where ${\left[{x}\right]}$ is a floor function.

You see that they are infinitude, but they behave differently: first is always greater 0, second is always less than 0, third alternates sign.

So, first sequence approaches $+\infty$, second sequence approaches $-\infty$, as for the third sequence we can't say what value it approaches.

Example 2. Sequence ${x}_{{n}}={{Q}}^{{n}}$ where ${\left|{Q}\right|}>{1}$ is also infinitude.

Indeed, ${\left|{x}_{{n}}\right|}={\left|{{Q}}^{{n}}\right|}>{E}$ when ${n}\cdot{\lg}{\left|{Q}\right|}>{\lg{{\left({E}\right)}}}$ or ${n}>\frac{{{\lg{{\left({E}\right)}}}}}{{{\lg}{\left|{Q}\right|}}}$.

So, we can take ${N}_{{E}}={\left[\frac{{{\lg{{\left({E}\right)}}}}}{{{\lg}{\left|{Q}\right|}}}\right]}$.

There is a connection between infinitesimal and infinitude:

Fact. If sequence ${x}_{{n}}$ is infinitude, then sequence $\alpha_{{n}}=\frac{{1}}{{x}_{{n}}}$ is infinitesimal, and vice versa.