More on Sequences....?

Question

People, answer ONLY if you know what you are talking about...otherwise don't bother!

Okay so basically, find a sequence with the general term a_n such that for any e>0, there exists x, y (not equal to each other) such that |a_x - a_y|
One such example, that I know if is a_n=ln(n). Any others out there?

Answer 1

You can get a fairly large class of sequences out of this - for instance, you could have a_n={1/n if n is even; 2^n if n is odd} - in fact any sequence made by alternating the terms of a convergent sequence with some sequence increasing without bound, or othwerwise not converging. Also, any sequnece corresponding to the partial sums on nonconvergent infinite series whose terms tend to zero. A whole bunch of stuff that you can just make up on the spot if you like.

Edit: if you saw my previous response, sorry about not reading the question carefully the first time. I'm kind of tired right now.

Answer 2

There are plenty of such sequences. Take any sequence b_n converging to 0 with the sum b_n diverging. Let a_n be the partial sums of the series sum b_n. Then a_(n+1)-a_n =b_(n+1) goes to 0 so your condition is satisfied.

Ex: b_n =1/n
b_n =1/[n*ln(n)]

Another way:
Let a_n=1 if n is even and 0 if n is odd. Clearly the sequence a_n diverges, but a_(n+2) -a_n =0 for all n.

Answer 3

The p-series \sum_{n=1}^{\infty} 1/n^p converges if p>1 and diverges if 0