Answer to Question #103846 in Calculus for BIVEK SAH

Question #103846
check whether the set [−3,7 [∩]− 7,3
]is a neighbourhood of 2 or not.
1
Expert's answer
2020-02-27T10:29:56-0500

First, let's visualize the intersection [3;7[[7;3[[-3;7[\cup[-7;3[ of given sets


Conclusion,


[3;7[[7;3[=[3;3[\boxed{[-3;7[\cup[-7;3[=[-3;3[}

Definition. Let aa be a point in R\mathbb{R} and let ε>0\varepsilon >0. The open inerval (aε;a+ε)\left(a-\varepsilon;a+\varepsilon\right) centered at aa is called the ε\varepsilon- NEIGHBOURHOOD of aa and is denoted Jε(a)J_\varepsilon(a). Notice that this neighbourhood consists of all numbers xx whose distance from aa is less than ε\varepsilon; that is, such that xa<ε|x-a|<\varepsilon.

(From John M. Erdman. A Problems Based Course in Advanced Calculus)


As can be seen from the definition, the neighborhood must be symmetric with respect to the point. And in our case, the given set is not symmetrical with respect to the point x = 2, since


ρ(2;3)=32=1<5=32=ρ(3;2)\rho(2;3)=|3-2|=1<5=|-3-2|=\rho(-3;2)



Conclusion,


[3;3[    is not a neighborhood of a point  x=2\boxed{[-3;3[ \,\,-\,\,\text{is not a neighborhood of a point}\,\,x=2}


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Assignment Expert
15.04.20, 16:08

Dear Pappu Kumar Gupta, please use the panel for submitting new questions.

Assignment Expert
15.04.20, 16:06

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Pappu Kumar Gupta
15.04.20, 07:53

Check whether the sequence, {Sn }, where Sn= 1/1!+1/3!+1/5!+....+1/(2n-1)! , is convergent or not , 2n 1 ! 1 5! 1 3! 1 1! 1 Sn − = + + +L+ Is convergent or not.

Pappu Kumar Gupta
15.04.20, 07:42

helpfull

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