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 `|x_n|>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 `+oo` or `-oo` and write: `lim x_n=+oo,lim_(n->+oo)x_n=+oo,x_n->+oo` or `lim x_n=-oo,lim_(n->+oo)x_n=-oo,x_n->-oo`. Also we say that sequence has infinite limit.

We already wrote that numbers `+-oo` 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 `oo-oo`.

Also, `+oo` is often written as `oo`.

Example 1. Consider sequences `x_n=n`, `x_n=-n`, `x_n=(-1)^(n+1)n`.

Corresponding lists are

`{1,2,3,4,...}`,

`{-1,-2,-3,-4,...}`,

`{1,-2,3,-4,...}`.

All variants are infinitude because `|x_n|=|n|>E` when `n>E`. Therefore, we can take `N_E>[E]`, where `[x]` 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 `+oo`, second sequence approaches `-oo`, as for the third sequence we can't say what value it approaches.

Example 2. Sequence `x_n=Q^n` where `|Q|>1` is also infinitude.

Indeed, `|x_n|=|Q^n|>E` when `n*lg|Q|>lg(E)` or `n>(lg(E))/(lg|Q|)`.

So, we can take `N_E=[(lg(E))/(lg|Q|)]`.

There is a connection between infinitesimal and infinitude:

Fact. If sequence `x_n` is infinitude, then sequence `alpha_n=1/x_n` is infinitesimal, and vice versa.