# Tagged Questions

For questions concerning the definition and properties of limit superior and limit inferior.

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### Finding lim sup and lim inf

Given the sequence $$(a_n)=\begin{cases} 3^{-n}, & \text{for even }n \\ 5^{-n}, & \text{for odd } n \end{cases}$$ How to find: $$\limsup_{n\to\infty}\frac{a_{n+1}}{a_n}$$ ...
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### Certain limit implies Cauchy sequence

I am reading a proof in which the following result is assumed. Let $(x_n)_{n\in\mathbb{N}}$ be a real valued sequence such that $$\limsup_{m\rightarrow\infty}\frac{1}{m}\log|x_{m+1}-x_m| < 0.$$ ...
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### If $\liminf\, |a_n|=0.$ Does there exists a subsequence of $\{a_n\}$ which has finite sum? [duplicate]

If $\liminf\, |a_n|=0.$ Does there exists a subsequence of $\{a_n\}$ which has finite sum? I tried to prove as follows: Since $\liminf\, |a_n|=0,$ then we can find $n_1<n_2<n_3\ldots$ such ...
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### Why does splitting up not work with this $\limsup$?

I recently came across this question here and there is something I don't understand. In the question we are considering a sequence $a_n$ with $\lim \sup_{n \to \infty}a_n \le \rho$ and we want to ...
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### Limit of the mean value of partial sums?

I'm supposed to construct a sequence of real numbers ${a_{n}}$ such that $\limsup_{n\to\infty} a_n=\infty$ but $\lim_{n\to\infty} b_n= \frac {a_{1}+a_{2}+...+a_{n}}{n} =0$.
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### If liminf exists, is the sequence is bounded below?

Let $\{a_n\}$ ($n \in {\mathbb Z}_+$ and $a_n \in {\mathbb R}$) be a sequence and \begin{align} \liminf_{n\to \infty} a_n > -\infty. \end{align} Does it mean $\{a_n\}$ is bounded below with a ...
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### Proving that x $\in \overline{\lim}(A_{n}$ implies for each N there exists at least one n$\geq$N

Letting {A$_{n}$} be a sequence of sets. We are given the definition $$\overline{\lim}(A_{n})= \cap_{n\geq 1}\cup_{k\geq n}A_{k}$$ The question wants us to prove that x $\in\overline{\lim}(A_{n})$ ...
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### Limit superior, limit inferior and a series involging $\sum_{k\nmid n}$k, where $1\leq k\leq n$

The purpose of this post is state assertions by the use of statements and hypothesis in an expository way and after I am asking for reasonable unconditionally results that you can provide us. Using ...
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### Problem involving limsup.

I had a problem while computing the limit superior below: If $r\sqrt[\leftroot{-2}\uproot{2}n]{|a_n|}\leq \sqrt[\leftroot{-2}\uproot{2}n]{M}$ $~~~~~~\forall n$ letting $n\rightarrow \infty$ ...
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### liminf of sequence of iid random variables

if $P(\liminf (X_n>a))=0$, does that mean $\liminf X_n <a$ almost surely? Where $X_n$ are iid random variables.
### Finding the $\lim \sup \cos (2\pi nx)$ for irrational $x$ in $[0,1]$
Let $x$ be an irrational number in $[0, 1]$. Find $\lim \sup \cos (2\pi nx)$ What I think I need to do is in some way take an $\epsilon$ neighborhood around some arbitrary point close to $1$ and ...
From Principles of Mathematical Analysis, third edition. p79, problem 4. Find the upper and lower limits of the sequence $\{s_n\}$ defined by s_1 = 0; s_{2m}=\frac{s_{2m-1}}{2}; s_{2m+1}= ...