# Infinitely Large Sequence

## Related Calculator: Limit Calculator

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.