query: find $\limsup$ and $\liminf$ of $a_n=\{\sqrt{n} - \lfloor\sqrt{n}\rfloor$ $n\in\mathbb{N}\}$
positive_passages.docid: 143819
positive_passages.text: Supremum of $\sqrt{n} -\left\lfloor\sqrt{n}\right\rfloor $
negative_passages.docid: 27844
negative_passages.text: How do we prove that $\lfloor0.999\cdots\rfloor = \lfloor 1 \rfloor$?
negative_passages.docid: 4024
negative_passages.text: Finding the limit of $\frac {n}{\sqrt[n]{n!}}$
negative_passages.docid: 38860
negative_passages.text: $P(\limsup A_n)=1 $ if $\forall A \in \mathfrak{F}$ s.t. $\sum_{n=1}^{\infty} P(A \cap A_n) = \infty$
negative_passages.docid: 116966
negative_passages.text: Prove $\limsup_{n \rightarrow \infty} a_n\geq1$ if $\lim_{n \rightarrow \infty} a_na_{n+1}=1$
negative_passages.docid: 19823
negative_passages.text: Inequality involving $\limsup$ and $\liminf$: $ \liminf(a_{n+1}/a_n) \le \liminf((a_n)^{(1/n)}) \le \limsup((a_n)^{(1/n)}) \le \limsup(a_{n+1}/a_n)$
negative_passages.docid: 101630
negative_passages.text: Limit $\lim_{n\to\infty}\sqrt n(\sqrt[n]x-1)$
negative_passages.docid: 91759
negative_passages.text: Evaluate $\lim_{x \to 0} (x\lfloor\frac{1}{x}\rfloor)$
negative_passages.docid: 25206
negative_passages.text: How to prove that $\lim\limits_{n\to \infty}a_{n+1}=\lim\limits_{n\to \infty}a_{n}$
negative_passages.docid: 44507
negative_passages.text: How can I find $\lim_{n\to \infty} a_n$
negative_passages.docid: 107200
negative_passages.text: Find $\lim_{n\to\infty}1+\sqrt[2]{2+\sqrt[3]{3+\sqrt[4]{4+\ldots+\sqrt[n]{n}}}}$
negative_passages.docid: 66824
negative_passages.text: Two definitions of $\limsup$
negative_passages.docid: 17534
negative_passages.text: Find $\lim_{n\to \infty}({1\over \sqrt{n^2+1}}+{1\over \sqrt{n^2+2}}+\cdots+{1\over \sqrt{n^2+n}})$
negative_passages.docid: 100733
negative_passages.text: $a_{n+1}=\log(1+a_n),~a_1>0$. Then find $\lim_{n \rightarrow \infty} n \cdot a_n$
negative_passages.docid: 59162
negative_passages.text: Prove $\liminf\limits_{n\to\infty} F(x_n)\le F(x)$
negative_passages.docid: 3974
negative_passages.text: convergence of $\sum_{n=1}^{\infty }\frac{(-1)^{[\sqrt{n}]}}{n}$ where $[a]$ means floor function of $a$
negative_passages.docid: 116811
negative_passages.text: Proofs with limit superior and limit inferior: $\liminf a_n \leq \limsup a_n$