Answer to Question #158002 in Calculus for Vishal

Question #158002

Use the definition of limit to prove that both of the sequences { 1 √ n √ } and {(−1)n n } converges to 0.


1
Expert's answer
2021-01-26T04:18:28-0500

Let an=1na_{n}=\frac{1}{\sqrt{n}} . Let ϵ>0\epsilon>0 is given. Then there exist some N>0 satisfying N>1ϵ2N>\frac{1}{\epsilon^{2}}.( Archimedean property of real numbers)


Therefore \forall nNn\geq N,


an0=1n=1n1N<ϵ|a_{n}-0|=|\frac{1}{\sqrt{n}}|=\frac{1}{\sqrt{n}}\leq \frac{1}{\sqrt{N}}<\epsilon .


Hence from the definition of limits an0.a_{n} \rightarrow 0.


Let bn=(1)nn.b_{n}= \frac{(-1)^{n}}{n}. Then nN\forall n\geq N,


bn0=(1)nn=1n1N<ϵ|b_{n}-0|=|\frac{(-1)^{n}}{n}|=\frac{1}{n}\leq \frac{1}{N}<\epsilon .


Hence from the definition of limits bn0.b_{n}\rightarrow 0.



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