# I have that the sequence $a_n=\{2-(-1)^n\}$ not converges.

1. Dec 21, 2011

### 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?

2. Dec 21, 2011

### Stimpon

Re: convergence

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

Last edited: Dec 21, 2011
3. Dec 21, 2011

### HallsofIvy

Staff Emeritus
Re: convergence

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.

4. Dec 21, 2011

### solakis

Re: convergence

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

5. Dec 21, 2011

### Deveno

Re: convergence

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.