1. ## converge or diverge?

Consider the sequence {a_n} (n is an element of N) given by a_n = (-1)^n. Does the sequence converge or diverge? Prove this.

2. Originally Posted by lindsays
Consider the sequence {a_n} (n is an element of N) given by a_n = (-1)^n. Does the sequence converge or diverge? Prove this.

Hint - Write down the sequence
-1,1,-1,1,-1,1,-1,1,....

This will go on and on...

So what will you conclude?

To prove it formally, Assume a limit L, prove both a_k and a_k+1 can be <1/2 (or a smaller number) (hence this becomes your epsilon) 'away' from L. Hence a contradiction to the definition of the limit

3. is it possible to prove this using the ratio test?

4. Originally Posted by lindsays
is it possible to prove this using the ratio test?
Just look at the terms of the sequence.
If a sequence converges, the almost all of its terms are very "close together".
If possibility true of this sequence?

5. Originally Posted by lindsays
is it possible to prove this using the ratio test?
The ratio test won't do you any good because it yields $\displaystyle 1$, which is inconclusive.

The definition of convergence of a sequence is:

A sequence $\displaystyle \{a_n\}$ converges to a point $\displaystyle a$ iff $\displaystyle \forall~\epsilon>0$, $\displaystyle \exists~M$ such that $\displaystyle m>M$ implies $\displaystyle |a_m-a|<\epsilon$.

Negating this statement gives us:

A sequence $\displaystyle \{a_n\}$ does not converge iff $\displaystyle \exists~\epsilon>0$ such that $\displaystyle \forall~M$, $\displaystyle \exists~ k,m>M$ such that $\displaystyle |a_m-a_k|\geq\epsilon$

Take $\displaystyle \epsilon=1$ and see what happens.