SPLIT - Convergence of sequence (alternating)

[Melind]

Let Xn = (−1)n for all n ∈ N. Show that the sequence (Xn) does not converge.
Can someone please help

 

[Deleted member 4993]

Let Xn = (−1)n for all n ∈ N. Show that the sequence (Xn) does not converge.
Can someone please help

Please start a new thread with a new problem

You have an alternating sequence.

Please show us what you have tried and exactly where you are stuck.

Please follow the rules of posting in this forum, as enunciated at:

READ BEFORE POSTING​

Please share your work/thoughts about this problem

 

[HallsofIvy]

I presume you mean (-1)^n or \(\displaystyle (-1)^n\).

It is very easy to write out as many terms of the sequence as you want:
\(\displaystyle a_1= (-1)^1= -1\)
\(\displaystyle a_2= (-1)^2= 1\)
\(\displaystyle a_3= (-1)^3= -1\)
\(\displaystyle a_4= (-1)^4= 1\)

So what do you think \(\displaystyle a_{100000}\) is? What about \(\displaystyle a_{100001}\)?

Now what is your understanding of what "a sequence converges" means?

 

[pka]

Let Xn = (−1)n for all n ∈ N. Show that the sequence (Xn) does not converge.
Can someone please help

The statement that \(\left(x_n\right)\to L\) means that for every for every \(\varepsilon>0\) there exist positive integer \(N\) such that if \(n\ge N\) such that \(\left|x_n-L\right|<\varepsilon\)
Now consider your sequence: \(\left|x_{n+1}-x_n\right|=\left|(-1)^{n+1}-(-1)^n\right|=\left|-1\right|\left|1-(-1)\right|=2\).
Now what?