# Proving Divergence

• Mar 30th 2012, 11:01 AM
renolovexoxo
Proving Divergence
The definition of a convergent squence is:
For every L in Real numbers, and all E>0 such that there exists N in Natural numbers n>/= N implies |xn-L|>E.
Negate this statement then use the negation to prove that the sequence xn=(-1)^n is not convergent.

I got the negation I think, but I'm not sure how to begin with the proof. We went over one example in class and it was for proving something converges. I got the negation as :
For all L in real numbers, there exists E>0 such that there exits N in natural numbers and n>/=N implies |xn-L|<E
• Mar 30th 2012, 11:38 AM
Sylvia104
Re: Proving divergence
Your definition of convergence is not correct. It should be

There exists L in Real numbers such that for all E>0 such that there exists N in Natural numbers such that n>/= N implies |xn-L|< E.

The negation of the definition of convergence should be: For every real number $L$ there exists a real number $\varepsilon>0$ such that for every natural number $N,$ there is a natural number $n\geqslant N$ such that $\left|x_n-L\right|\geqslant\varepsilon.$ Use this to prove the divergence of $\left(x_n\right)_{n=1}^\infty$ where $x_n=(-1)^n.$ Hint: Given $L,$ let $\varepsilon=\min\left\{|L+1|,|L-1|\right\}$ if $L\ne\pm1$ and $\varepsilon=2$ if $L=\pm1.$
• Mar 31st 2012, 03:05 PM
renolovexoxo
Re: Proving Divergence
I don't understand how you are proving the negation. We just used the pieces of the definition to prove convergence in class. Is that what you are doing here?
• Mar 31st 2012, 04:01 PM
Plato
Re: Proving Divergence
Quote:

Originally Posted by renolovexoxo
I don't understand how you are proving the negation. We just used the pieces of the definition to prove convergence in class. Is that what you are doing here?

Is this true: $\left( {\forall n} \right)\left[ {\left| {{x_{n + 1}} - {x_n}} \right| = 2} \right]~?$
If that is true, how can the terms of the sequence ${\left( { - 1} \right)^n}$ get "close together"?
• Mar 31st 2012, 05:32 PM
renolovexoxo
Re: Proving Divergence
I'm getting lost. I feel like there is something there that I'm not understanding.
• Mar 31st 2012, 06:09 PM
Plato
Re: Proving Divergence
Quote:

Originally Posted by renolovexoxo
I'm getting lost. I feel like there is something there that I'm not understanding.

I think that you need a live-sit-down with an instructor/lecturer.
We here just not equipped to aid with this kind profound confusion.
Talk to your instructor!
• Mar 31st 2012, 07:23 PM
renolovexoxo
Re: Proving Divergence
I can't or I would. Thanks for trying I suppose. I understand the second piece, but between the first response and the first line of what you wrote I can't figure it out. Do I only need the fact that they cannot equal 2 or get closer together for all n? That's really where I'm getting hung up. If asked to explain it, I could, it's more where I am supposed to get this fact from that is causing an issue. Should I just have been able to know that?