Divergent sequence

Definition

A real sequence $(a_n)$ is divergent if it is not a convergent sequence. Thus, for each real number $L$, the sequence $(a_n)$ does not converge to $L$.

Remarks

The sequence $a_n = (-1)^n$ is divergent, and it is bounded. The sequence $a_n = n$ is divergent, and it tends to infinity. Some texts say that a sequence that tends to $+\infty$ diverges to $+\infty$. It is still divergent in the sense of this definition.

Explanations

L1 human

A divergent sequence has no limit. Its terms can grow without bound, as in $1, 2, 3, \dots$, or they can jump between values, as in $1, -1, 1, -1, \dots$.