Limit Superior and Limit Inferior

For sequences that don't converge, we can still extract information about their limiting behavior using limit superior ( $\limsup$ ) and limit inferior ( $\liminf$ ). These measure the "eventual maximum" and "eventual minimum" behavior of a sequence. Every bounded sequence has a $\limsup$ and $\liminf$ , even if it doesn't have a limit.

Definitions

Definition2.1Limit superior and limit inferior

Let $(a_n)$ be a bounded sequence in $\mathbb{R}$ . Define:

$\limsup_{n \to \infty} a_n = \lim_{n \to \infty} \sup\{a_k \mid k \geq n\} = \lim_{n \to \infty} \sup_{k \geq n} a_k,$

$\liminf_{n \to \infty} a_n = \lim_{n \to \infty} \inf\{a_k \mid k \geq n\} = \lim_{n \to \infty} \inf_{k \geq n} a_k.$

RemarkIntuition

$\sup_{k \geq n} a_k$ is the supremum of the "tail" starting at $n$ . As $n$ increases, this supremum can only decrease (or stay constant), so the limit exists.
Similarly, $\inf_{k \geq n} a_k$ increases as $n$ increases, so its limit exists.
$\limsup a_n$ captures the largest value the sequence "clusters near" infinitely often.
$\liminf a_n$ captures the smallest value the sequence clusters near infinitely often.

ExampleLimsup and liminf of (-1)^n

Let $a_n = (-1)^n$ . Then:

$\sup_{k \geq n} a_k = 1$ for all $n$ (since the sequence keeps hitting $1$ ).
$\inf_{k \geq n} a_k = -1$ for all $n$ (since it keeps hitting $-1$ ).

Thus $\limsup a_n = 1$ and $\liminf a_n = -1$ . The sequence does not converge, and $\limsup \neq \liminf$ .

ExampleConvergent sequence

If $a_n \to L$ , then $\limsup a_n = \liminf a_n = L$ . Indeed, for large $n$ , all terms lie in $(L - \epsilon, L + \epsilon)$ , so

$\sup_{k \geq n} a_k \to L \quad \text{and} \quad \inf_{k \geq n} a_k \to L.$

ExampleLimsup of 1 + (-1)^n/n

Let $a_n = 1 + (-1)^n/n$ . Then:

For even $n = 2m$ , $a_n = 1 + 1/(2m) \to 1$ from above.
For odd $n = 2m+1$ , $a_n = 1 - 1/(2m+1) \to 1$ from below.

So $\limsup a_n = \liminf a_n = 1$ , and $a_n \to 1$ .

Properties

Theorem2.1Basic properties of limsup and liminf

Let $(a_n)$ and $(b_n)$ be bounded sequences. Then:

$\liminf a_n \leq \limsup a_n$ .
$(a_n)$ converges to $L$ if and only if $\limsup a_n = \liminf a_n = L$ .
If $a_n \leq b_n$ for all $n$ , then $\limsup a_n \leq \limsup b_n$ and $\liminf a_n \leq \liminf b_n$ .
$\limsup(a_n + b_n) \leq \limsup a_n + \limsup b_n$ (subadditivity).
$\liminf(a_n + b_n) \geq \liminf a_n + \liminf b_n$ .

RemarkProof of (2)

If $\limsup a_n = \liminf a_n = L$ , then $\inf_{k \geq n} a_k \leq a_n \leq \sup_{k \geq n} a_k$ . Taking $n \to \infty$ , by the squeeze theorem, $a_n \to L$ .

Conversely, if $a_n \to L$ , then for large $n$ , all tails are bounded by $L \pm \epsilon$ , so $\sup_{k \geq n} a_k \to L$ and $\inf_{k \geq n} a_k \to L$ .

ExampleSubadditivity of limsup

Let $a_n = (-1)^n$ and $b_n = -(-1)^n$ . Then $a_n + b_n = 0$ for all $n$ , so

$\limsup(a_n + b_n) = 0.$

But $\limsup a_n = 1$ and $\limsup b_n = 1$ , so

$\limsup(a_n + b_n) = 0 < 1 + 1 = \limsup a_n + \limsup b_n.$

Thus the inequality in (4) can be strict.

Applications

ExampleRatio test for series

The ratio test for series $\sum a_n$ states: if

$\limsup_{n \to \infty} \left|\frac{a_{n+1}}{a_n}\right| < 1,$

then $\sum a_n$ converges absolutely. This uses $\limsup$ to handle sequences that don't have limits (e.g., alternating terms).

ExampleRadius of convergence

For a power series $\sum a_n x^n$ , the radius of convergence is

$R = \frac{1}{\limsup_{n \to \infty} |a_n|^{1/n}}.$

The use of $\limsup$ (rather than $\lim$ ) makes the formula valid even when $|a_n|^{1/n}$ does not converge.

Summary

Limit superior and limit inferior provide robust notions of limiting behavior:

They exist for all bounded sequences, even non-convergent ones.
A sequence converges iff $\limsup = \liminf$ .
They satisfy subadditivity and monotonicity properties.
Applications include the ratio test and radius of convergence for series.

See Monotone Convergence for convergence of monotone sequences, and Series of Functions for applications to power series.