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.