Mathematik-Online problems: Problem 104: Convergence of Series, Multiple Choice

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	Mathematik-Online problems:
	Problem 104: Convergence of Series, Multiple Choice

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

Given the real sequences

and

, as well as the series

$\displaystyle A=\sum_{n=1}^\infty a_n, \quad B=\sum_{n=1}^\infty b_n \quad {\mbox{and}} \quad C=\sum_{n=1}^\infty a_nb_n.$

Mark which statements are always true respectively false, and give reasons for your answers.

$\sqrt[n]{n^{1-n}}\leq a_n\leq \sqrt[n]{n^{1-n/2}}\,, \ \forall\, n\in\mathbb{N} \ \Longrightarrow \ (a_n)$ is convergent true $\bigcirc$ false $\bigcirc$

$a_{n+1}/a_n$ is strictly monotonic increasing $\Longrightarrow$ is divergent true $\bigcirc$ false $\bigcirc$

$\sqrt[n]{\vert a_n\vert}<1, \ \forall\, n\in\mathbb{N} \ \Longrightarrow$ is convergent true $\bigcirc$ false $\bigcirc$

${\displaystyle{\lim_{n\to\infty}}} n\sqrt{n}\,a_n = 2\pi \ \Longrightarrow$ ist absolute convergent true $\bigcirc$ false $\bigcirc$

and are convergent $\Longrightarrow$ is convergent true $\bigcirc$ false $\bigcirc$

(Authors: Apprich/Höfert)

Solution:

Lösung (german)

[Links]

automatisch erstellt am 12. 4. 2005

$\sqrt[n]{n^{1-n}}\leq a_n\leq \sqrt[n]{n^{1-n/2}}\,, \ \forall\, n\in\mathbb{N} \ \Longrightarrow \ (a_n)$ is convergent	true $\bigcirc$	false $\bigcirc$
$a_{n+1}/a_n$ is strictly monotonic increasing $\Longrightarrow$ is divergent	true $\bigcirc$	false $\bigcirc$
$\sqrt[n]{\vert a_n\vert}<1, \ \forall\, n\in\mathbb{N} \ \Longrightarrow$ is convergent	true $\bigcirc$	false $\bigcirc$
${\displaystyle{\lim_{n\to\infty}}} n\sqrt{n}\,a_n = 2\pi \ \Longrightarrow$ ist absolute convergent	true $\bigcirc$	false $\bigcirc$
and are convergent $\Longrightarrow$ is convergent	true $\bigcirc$	false $\bigcirc$