# Questions tagged [limsup-and-liminf]

For questions concerning the definition and properties of limit superior and limit inferior of sequences of sets or real numbers.

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### Coinciding Limsup and limit

Let $(x_n)_{n \geq 1}$ be a sequence of real numbers. Suppose that we are able to show that for a fixed number $m$, $(y_n)_{n \geq 1}:= (x_{n+m})$ and we know that $\lim_{n\to\infty}(y_n)=x$ for some ...
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### An inequality for fixed points of dynamical systems

Suppose $f:X\to X$ is a continuous map such that each iterate has finitely many fixed points. Define $h(f) = \limsup_{k\to \infty} \frac{\log \#Fix(f^k)}{k}$ where $\#Fix(f^k)$ is the number of fixed ...
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### An exercise on counting fixed points from Milnor

I am working on the following exercise (Problem 6-b, pg. 6-22) from Milnor's Dynamical Systems notes: Problem 6-b Let $f$ be any self map such that each iterate has only finitely many fixed points. ...
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1 vote
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### Find $a, b \in \mathbb{R}$ such that power series $\sum_{n=1}^{\infty} \frac{\arctan n^a}{n^b} x^n$ converges.

Find $a, b \in \mathbb{R}$ such that power series $$\sum_{n=1}^{\infty} \frac{\arctan n^a}{n^b} x^n$$ converges. I had problem in finding radius of convergence, more precisely I don't know how to ...
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### Find $\limsup (a_n)^{\frac{1}{n}}$ given that $\limsup a_n =R>0$, and $a_n\geq 0$ for each $n\in\mathbb{N}$

I'm trying to find if there's a relation between $\limsup (a_n)^{\frac{1}{n}}$ and $\limsup a_n$, given that $\limsup a_n =R>0$ and $a_n\geq 0$ for each $n\in\mathbb{N}$. I've found some examples ...
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### Limit superior and limit inferior of sets

I have searched for the answer in wikipedia and math stackexchange. However, I do not have any background in real analysis and all the answers seem very complicated to me to understand. I am wondering ...
1 vote
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### How to estimate numerically a lim sup/lim inf

Suppose I have a sequence of random variables $X_n$ s.t. $\lim_{n\to\infty}X_n=c$ a.s. For example, we could take the $X_n$ to be the sample mean from some well-behaved distribution. If I have a way ...
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### Minimal assumption on points of discontinuity of $f$ so that $\liminf_{\epsilon \to \infty} \frac{f(x)}{f(x+\frac{t}{\epsilon} )}=1$

Suppose we are given a function $f(x)$. We want to show the following claim: \begin{align} \liminf_{\epsilon \to \infty} \frac{f(x)}{f(x+\frac{t}{\epsilon} )}=1, \end{align} almost everywhere $(x,t)$ ...
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### Set of points where lower left Dini derivative dominates the upper right Dini derivative is countable

I am self studying analysis and came across a theorem that states that for any real valued function $f$, the set of points where the upper right derivative of $f$ is strictly less than the lower left ...
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### Is $\liminf f(x)$ always unique?

Searching for the question doesn't seem to give any relevant answers on stack exchange or on google. I know that $\liminf_{n \rightarrow \infty} f(x_n)$ is always unique, but I haven't seen a proof ...
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### Understanding what $\displaystyle \lim\sup_{x \to a, x \in E} f(x)$ means

I recently learned of the following extension to the definition of a limit: Let $S \subset \mathbb{R}$, let $f: S \rightarrow \mathbb{R}$, let $a$ be a limit point of $S$, and let $L \in \mathbb{R}$. ...
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