Show that if then the sequence { } converges, but not conversely.

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- Sep 11th 2008, 11:20 AMjohnny0987Series and Sequence Covergence
Show that if then the sequence { } converges, but not conversely.

- Sep 11th 2008, 12:05 PMLaurent
Notice that , so that converges if, and only if is a convergent series.

If a series is absolutely convergent, then it converges. However the converse is not true. Using a counterexample of this property (like , for instance), you can deduce a sequence which converges, but such that diverges. - Sep 11th 2008, 12:25 PMbkarpuz
Let

then we know that

provided that is a nonnegative sequence of reals.

From your question, we see that

holds, i.e. there exists and such that for all , this implies that is a Cauchy sequence (see the definition of Cauchy sequences and let ), and Cauch sequences are always convergent in complete spaces ( is complete).

For the converse part, it suffices to give a counter example.

Let

Then, we see that

And therefore, we get

.............................

.............................

On the other hand, we have .

This completes the proof. - Sep 11th 2008, 02:04 PMLaurent
- Sep 11th 2008, 08:48 PMbkarpuz