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.

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- October 10th 2009, 11:39 AMlindsaysconverge 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.

- October 10th 2009, 11:54 AMaman_cc

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 - October 10th 2009, 12:28 PMlindsays
is it possible to prove this using the ratio test?

- October 10th 2009, 12:32 PMPlato
- October 10th 2009, 08:51 PMredsoxfan325
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.