Mathematik-Online problems: Problem 91: Convergence ans Limits of Sequences

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	Mathematik-Online problems:
	Problem 91: Convergence ans Limits of Sequences

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

Analyse if the given sequences are convergent and give the limit for each convergent sequence.

a) $a_n={\displaystyle{\frac{(3n-4)(n^2+1)}{7n\,(2n^2+10000)}}}$ b) $a_n={\displaystyle{n\left(1-\sqrt{1-\frac{c}{n}}\; \right),}} \quad c\leq 1$

c) $a_n={\displaystyle{\left(1-\frac{c}{n}-\frac{c^2}{n^3}\;\right)^{\!n},}} \quad c\in\mathbb{R}$ d) $a_n={\displaystyle{{\mathrm{sp}}(A^n), \quad A=\left(\begin{array}{cc} 0 & {\mathrm{i}} \\ {\mathrm{i}} & 0\end{array}\right)}}$

e) $a_n={\rm {e}}^{-\frac{1}{3}n\pi}$ f) $a_n={\rm {e}}^{-\frac{1}{3}n\pi{\mathrm{i}}}$

(Authors: Kirchgässner/Werner/Hesse/Apprich/Höfert)

Solution:

Lösung (german)

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automatisch erstellt am 13. 12. 2007

a)	$a_n={\displaystyle{\frac{(3n-4)(n^2+1)}{7n\,(2n^2+10000)}}}$	b)	$a_n={\displaystyle{n\left(1-\sqrt{1-\frac{c}{n}}\; \right),}} \quad c\leq 1$
c)	$a_n={\displaystyle{\left(1-\frac{c}{n}-\frac{c^2}{n^3}\;\right)^{\!n},}} \quad c\in\mathbb{R}$	d)	$a_n={\displaystyle{{\mathrm{sp}}(A^n), \quad A=\left(\begin{array}{cc} 0 & {\mathrm{i}} \\ {\mathrm{i}} & 0\end{array}\right)}}$
e)	$a_n={\rm {e}}^{-\frac{1}{3}n\pi}$	f)	$a_n={\rm {e}}^{-\frac{1}{3}n\pi{\mathrm{i}}}$