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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 $ (a_n)$ and $ (b_n)$, 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$ $ A$ is divergent  true $ \bigcirc $  false $ \bigcirc $
$ \sqrt[n]{\vert a_n\vert}<1, \ \forall\, n\in\mathbb{N} \ \Longrightarrow$ $ A$ is convergent  true $ \bigcirc $  false $ \bigcirc $
$ {\displaystyle{\lim_{n\to\infty}}} n\sqrt{n}\,a_n = 2\pi \ \Longrightarrow$ $ A$ ist absolute convergent  true $ \bigcirc $  false $ \bigcirc $
$ A$ and $ B$ are convergent $ \Longrightarrow$ $ C$ is convergent  true $ \bigcirc $  false $ \bigcirc $

(Authors: Apprich/Höfert)

Solution:


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  automatisch erstellt am 12.  4. 2005