What does $\lim \inf _ { n \rightarrow \infty }$ mean?

Question

I'm new into Mathematical Analysis, and my textbook says:

Consider the series: $\frac { 1 } { 2 } + \frac { 1 } { 3 } + \frac { 1 } { 2 ^ { 2 } } + \frac { 1 } { 3 ^ { 2 } } + \frac { 1 } { 2 ^ { 3 } } + \frac { 1 } { 3 ^ { 3 } } + \frac { 1 } { 2 ^ { 4 } } + \frac { 1 } { 3 ^ { 4 } } + \cdots$

for which

$\liminf _ { n \rightarrow \infty } \frac { a _ { n + 1 } } { a _ { n } } = \lim _ { n \rightarrow \infty } \left( \frac { 2 } { 3 } \right) ^ { n } = 0$

$\liminf _ { n \rightarrow \infty } \sqrt [ n ] { a _ { n } } = \lim _ { n \rightarrow \infty } \sqrt [ 2 n ] { \frac { 1 } { 3 ^ { n } } } = \frac { 1 } { \sqrt { 3 } }$

$\limsup _ { n \rightarrow \infty } \sqrt [ n ] { a _ { n } } = \lim _ { n \rightarrow \infty } \sqrt [ 2 n ] { \frac { 1 } { 2 ^ { n } } } = \frac { 1 } { \sqrt { 2 } }$

$\limsup _ { n \rightarrow \infty } \frac { a _ { n + 1 } } { a _ { n } } = \lim _ { n \rightarrow \infty } \frac { 1 } { 2 } \left( \frac { 3 } { 2 } \right) ^ { n } = + \infty$

My question is: What does $\lim \inf _ { n \rightarrow \infty }$ mean? And why $\lim \inf _ { n \rightarrow \infty } \frac { a _ { n + 1 } } { a _ { n > } } = \lim _ { n \rightarrow \infty } \left( \frac { 2 } { 3 } \right)$?

Answer 1

You can refer to your notes or to the wiki page Limit superior and limit inferior for the definition.

It is just a generalization for the limit of a sequence, in particular we consider

$$I_n=\inf\{a_n:k\ge n\}$$

and since $I_n$ is an increasing sequence the limit always exist (finite or infinite) and then we can consider and define

$$\liminf_{n\to \infty} a_n =\lim_{n\to \infty} I_n $$

To show that for a given sequence $a_n$ we have $\liminf_{n\to \infty} a_n =L$ we need to show two fact

• $a_n \ge b_n \quad b_n\to L$
• find a sub-sequence $a_{i_n}$ such that $a_{i_n}\to L$