Prove that $\lim\limits_{n->\infty}\sup a_n\le q\iff\forall\epsilon>0\: \exists N\in\mathbb{N}:n\ge N\:\forall n\in\mathbb{N}\implies a_n< q+\epsilon$

score 102 · Answer 6 · 2026-03-27 15:18:57

102

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Series of functions: convergence interval type vs. monotony of the function and calculation of $\mathrm{sup}_{x \in [r,\infty)}|f_n(x)|$

Published on 27 Mar 2026 - 15:18

score 227 · Answer 7 · 2016-10-22 18:46:05

227

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For $x_n>0$ prove that $\underline\lim\frac{x_{n+1}}{x_n}\le\underline\lim\sqrt[n]{x_n}$ etc...

Published on 22 Oct 2016 - 18:46

score 366 · Answer 8 · 2016-10-23 04:32:08

366

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Is there a way to relate lim sup/inf of sets to the definition for sequences?

Published on 23 Oct 2016 - 4:32

score 51 · Answer 9 · 2016-10-29 07:40:34

51

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Basic real analysis - Root test (question on example from book)

Published on 29 Oct 2016 - 7:40

score 299 · Answer 10 · 2016-10-30 04:09:41

299

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Basic real analysis - Proving convergence of series

Published on 30 Oct 2016 - 4:09

score 1.1k · Answer 11 · 2016-10-30 12:26:10

1.1k

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Is it true that $\limsup \max (a_n, b_n) = \max(\limsup a_n, \limsup b_n)$?

Published on 30 Oct 2016 - 12:26

score 34 · Answer 12 · 2016-10-31 09:35:24

34

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$\{x_n\}$ is nonnegative. Show $\limsup_{n \to \infty} {something} \le \limsup_{n \to \infty} {another thing}$. Also give examples of $\{x_n\}$.

Published on 31 Oct 2016 - 9:35

score 40 · Answer 13 · 2016-10-31 20:28:36

40

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Let $(a_k)$, with $0<\liminf |a_k|\le\limsup |a_k|<\infty$. Determine the radius of convergence of $\sum a_kX^k$

Published on 31 Oct 2016 - 20:28

score 673 · Answer 14 · 2016-11-03 14:25:20

673

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Show that if $s_n$ and $t_n$ are two sequences in $\mathbb{R}$, then $\lim \sup (s_n + t_n) \leq \lim \sup (s_n) + \lim \sup (t_n)$

Published on 03 Nov 2016 - 14:25

score 178 · Answer 15 · 2016-11-04 11:44:18

178

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Prove that $\limsup_{n \to \infty} x_n \cdot y_n = x \cdot\limsup_{n \to \infty} y_n$

Published on 04 Nov 2016 - 11:44

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Prove that $\lim\limits_{n->\infty}\sup a_n\le q\iff\forall\epsilon>0\: \exists N\in\mathbb{N}:n\ge N\:\forall n\in\mathbb{N}\implies a_n< q+\epsilon$

Series of functions: convergence interval type vs. monotony of the function and calculation of $\mathrm{sup}_{x \in [r,\infty)}|f_n(x)|$

For $x_n>0$ prove that $\underline\lim\frac{x_{n+1}}{x_n}\le\underline\lim\sqrt[n]{x_n}$ etc...

Is there a way to relate lim sup/inf of sets to the definition for sequences?

Basic real analysis - Root test (question on example from book)

Basic real analysis - Proving convergence of series

Is it true that $\limsup \max (a_n, b_n) = \max(\limsup a_n, \limsup b_n)$?

$\{x_n\}$ is nonnegative. Show $\limsup_{n \to \infty} {something} \le \limsup_{n \to \infty} {another thing}$. Also give examples of $\{x_n\}$.

Let $(a_k)$, with $0<\liminf |a_k|\le\limsup |a_k|<\infty$. Determine the radius of convergence of $\sum a_kX^k$

Show that if $s_n$ and $t_n$ are two sequences in $\mathbb{R}$, then $\lim \sup (s_n + t_n) \leq \lim \sup (s_n) + \lim \sup (t_n)$

Prove that $\limsup_{n \to \infty} x_n \cdot y_n = x \cdot\limsup_{n \to \infty} y_n$

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