Typical limits of sequences

Some limits recur so often that they are useful as an immediate reference.

Example — Typical limits

• $a_n = 1/n \to 0$.
• $a_n = n \to +\infty$.
• $a_n = (-1)^n$: indeterminate (alternates $+1, -1$).
• $a_n = q^n$ (geometric progression): if $|q|<1$ it tends to $0$; if $q = 1$ it is constant $= 1$; if $q>1$ it tends to $+\infty$; if $q\le -1$ it is indeterminate.
• $a_n = n!/n^n \to 0$ (the factorial grows less than $n^n$).

The case $q^n$ is particularly important: the $n$-th power of the common ratio governs the behaviour of every geometric progression.

Links

Topics: Sequences
Concepts: Limit of a sequence · Geometric progression
Skills: Compute limits · Reason by cases