What are the applications / usages of the limes superior?

Question

I want to write an article about the limes superior. For motivating this concept I am looking for applications. Currently I just have found the root test and (connected to the root test) the Cauchy-Hadamard theorem. Are there other applications / theorems / concepts where the limes superior plays an important role? Thanks for your answers.

Answer 1

As mentioned in the comments, the root test is secretly an extremely important theorem. Although other tests for specific series are usually more useful, the root test can be used on the power series representation of a function, and then one can solve for $x$. In this manner, the root test makes it easy to compute the radius of convergence of the power series of any function.

Limit inferiors are used in continued fraction theory and rational approximation theory, where you have a sequence of approximations, but because we're only interested in the best ones the limit inferior of the error term is important.

This inequality is pretty useful in measure theory.

Checking out the relevant tag is another good way to see uses of the concept.

Answer 2

I recommend to take a look at the Wikipedia Page: Wikipedia

Limes superior (and limes inferior) play an important role in number theory as well. E.g. consider arithmetic functions which appear 'unpredictable' due to the prime factorization. Therefore various statements and theorems use limes superior.

To give you some 'famous' examples, consider the asymptotic growth rate of the divisor sum function $\sigma(n)$: $$ \limsup_{n\rightarrow \infty} \frac{\sigma(n)}{n \log\log n} = e^\gamma. $$

Or statements like $$ \limsup_{n\rightarrow \infty} \frac{\phi(n)}{n} =1 $$ or $$ \liminf_{n\rightarrow \infty} \phi(n) \frac{\ln\ln n}{n} = e^{-\gamma} $$ where $\phi$ is the Totient function.