Sequence tending to infinity

Definition

Let $(a_n)$ be a real sequence. The sequence tends to $+\infty$ if, for each real number $M$, there is a natural number $N$ such that $a_n > M$ for all $n \ge N$. Then the notation is $a_n \to +\infty$ as $n \to \infty$, or $\lim_{n \to \infty} a_n = +\infty$. The sequence tends to $-\infty$ if $-a_n \to +\infty$.

Remarks

The sequences $n$, $n^2$ and $2^n$ tend to $+\infty$. A sequence that tends to $+\infty$ is not bounded above, so it is a divergent sequence. The converse fails. The sequence $0, 1, 0, 2, 0, 3, \dots$ is not bounded above, but it does not tend to $+\infty$.

Explanations

L1 human

A sequence tends to infinity when its terms become larger than any number you name, and stay larger. It is not enough that the terms are sometimes large.