Tuesday, 17 June 2014

calculus - To prove a sequence is Cauchy




I have a sequence:
an=3+3+...3 , it repeats n-times.




and i have to prove that it is a Cauchy's sequence.
So i did this:
As one theorem says that every convergent sequence is also Cauchy, so i proved that it's bounded between 3 and 3 (with this one i am not sure, please check if i am right with this one.)And also i proved tat this sequence is monotonic. (with induction i proved this: anan+1
so if it's bounded and monotonic, therefore it is convergent and Cauchy.
I am just wondering if this already proved it or not? And also if the upper boundary - supremum if you wish - is chosen correctly.
I appreciate all the help i get.


Answer



an+1=an+3 a2n+1=an+3 as n a2n+1=an+3 x2=x+3 x2x3=0 andit convergents to the x=1+132


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