I have that the sequence [itex]a_n=\{2-(-1)^n\}[/itex] not converges.

[solakis]

I have that the sequence $a_n=\{2-(-1)^n\}$ not converges. I must show this with the rigorous definition.

I think use $\exists{\epsilon>0}\forall{N\in\mathbb{N}}\exists{n\geq N}:|a_n-\ell|\geq\epsilon$

How i can continue?

 

[Stimpon]

Try showing that it's not a cauchy sequence instead and then just say "therefore it is not convergent."

 

[HallsofIvy]

Alternatively, note that $a_n= 1$ for n even, $a_n= 3$ for n odd. For any l, there exist arbitarily large n such that $|a_n- l|> 1$, half the distance between 1 and 3.

 

[solakis]

Alternatively, note that $a_n= 1$ for n even, $a_n= 3$ for n odd. For any l, there exist arbitarily large n such that $|a_n- l|> 1$, half the distance between 1 and 3.

So which do you think should be the value of ε>ο ??

 

[Deveno]

So which do you think should be the value of ε>ο ??

HallsofIvy just told you what epsilon to use, half the value of the difference of the two possible values any term of the sequence can have.