The question is in the topic (EDIT: should be "sequence" instead of "series"). The [EDIT: sequence] is as follows:

{0,1,0,0,1,0,0,0,1,0,0,0,0,1,...}

No, this isn't homework (anymore). I'm just not satisfied with the "official" answer. I won't give the answer yet because I don't want it to influence the way people respond, if they do.

Thanks.

divergent

Okay, well, that's pretty much the answer. But can you show why? Also, I disagree because it seems that as n -> infinity, would there not eventually be an infinite number of 0s such that a 1 is never encountered again? Keep in mind that infinity is really, really big, in fact bigger than you think it is, and bigger still.

(someone feel free to correct me if im wrong)

a series can't converge to 0 if it is already 0. as you stated, there will eventually be an infinite amount of zeros, but if its 0 how can it converge to itself?

No. You will still encounter 1's, just not as often.

Given {an} = {0,1,0,0,1,0,0,0,1,0,0,0,0,1,...}, as n -> infinity, the number of 0s between a 1 and a subsequent 1 also -> infinity.

I mean, if lim(x -> infinity) 1 / x = 0, and lim(x -> infinity) x / x + 1 = 1, shouldn't lim(n -> infinity) {an} = 0?

It has been a while since I have done this so hopefuly I'm giving you the right answer. I think 7 had it right.


if I remember an infinate series is and infinate sum. so it is solved: by adding all the numbers in the series up. (if thats not what this is then let me know)

If it is a sum like I think I remember it than it would be divergent: Eventualy it will not get any bigger because you will be adding 0+0+0.... but untill that point you are adding 1's into the sum periodicly. so the sum is infinity by the time you get to 0+0+0.... so it's not getting any bigger but it has already added into the sum an infinate number of 1's before it gets to that point.


sorry if this makes no sense I'm not going on the worlds best memory.

If you write the sequence (not series) as a piecewise function describing how to write the 1s and 0s, you will see that as they go to infinity, they will grow w/o bound thus be divergent.

Okay, sorry, I misused the term. A series is a sum of a sequence. What I meant was that this was a sequence.

dalf ftw

The sequence doesn't converge either. (Not convinced that convergence is the correct term for what you're trying to describe, but I'll run with it cos I can't think of the term it SHOULD be).

If you show me a point in the sequence where you've got a run of zeroes, I can always show you a point further in the sequence that is a 1.

Proof by contradiction?