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
The ratio test won't do you any good because it yields , which is inconclusive.
The definition of convergence of a sequence is:
A sequence converges to a point iff , such that implies .
Negating this statement gives us:
A sequence does not converge iff such that , such that
Take and see what happens.