Bounds on infinite sequences with a known limit

[Nelphine]

Hello!

Unfortunately, I have not spent as much time as I should have on limits, or sequences, or their properties. In trying to work on a number theory math proof I have come across the following:

I have an infinite sequence of numbers, all between 0 and 1 inclusive. I know that the limit of this sequence is 0.4. I know that at some point, call it x, where x is the nth term in the sequence, all the numbers in the sequence past x get very close to the limit (in fact as close as I would like). For example, I know already that for n = 1 million, all the numbers past x will be within 0.001 of 0.4.

However, what I need to know is information about the numbers BEFORE my currently known x. So what I would like to know is if there are tests that can find a less precise bound on a sequence that has a known limit.

For instance, (if it were true) is there a test that would tell me that for n = 50, all the numbers in the sequence past x will be within 0.4 of 0.4? More specifically, are there tests that will tell me the most IMPRECISE bound on the sequence? (Aside from the given bound of between 0 and 1, since it's possible none of the numbers will ever actually be 0 or 1)(And then, I actually have an infinite series of such sequences, and none of them have quite the same properties, nor quite the same limits, although all the limits are known; so I assume it's possible I might need different tests depending on which particular sequence I was looking at.)

And for those that are curious, since limits are usually the important part, the numbers that actually contain the information I need are usually within the first 0.1% of the numbers that are before my currently known x's. (So for all the sequences I know an x such as the example I used at the start where n = 1 million; but all the important information for that particular sequence comes from the first 1000 numbers. (However, some of the sequences still have billions of important numbers, its simply that the known bound for the limit occurs at a VERY high n value.)

 

[johnqwertyful]

Simple, every finite set of real numbers has a greatest element.

So if n=a million, then you can just compare the first million terms to find the greatest element, thus a bound. If you're asking if just because a sequence converges, we can say anything about its greatest element, no.

For example:
1, 1, 1, ...
vs
4, 1, 1, ...
23423423423423432, 1, 1, ...
123123123.324234, 1, 1, ...

They all converge to 1, yet have wildly different maximums.

 

[Nelphine]

hum. that would actually be too imprecise. although it's obviously what I asked for, it's also unfortunately not at all what I want.

So. I want to find.. every n value where all numbers in the sequence past the nth term will be closer to the limit than the nth term. The x value I used in my example is one such; another such is simply the greatest element as posted by johnqwertyful, but since neither of those helps me, I need to find other ones.

Edit: If possible I would like to be able to do this without having to determine all the numbers in the sequence first; I'd like to be able to determine where these numbers will be, and then go and find what those numbers are based on which ones have that property.

 

[johnqwertyful]

hum. that would actually be too imprecise. although it's obviously what I asked for, it's also unfortunately not at all what I want.

So. I want to find.. every n value where all numbers in the sequence past the nth term will be closer to the limit than the nth term. The x value I used in my example is one such; another such is simply the greatest element as posted by johnqwertyful, but since neither of those helps me, I need to find other ones.

Edit: If possible I would like to be able to do this without having to determine all the numbers in the sequence first; I'd like to be able to determine where these numbers will be, and then go and find what those numbers are based on which ones have that property.

I don't understand this. Every sequence is wildly different.

Sn=1/n for n≠10
Sn=5 for n=10

Tn=1/n for n≠40
Tn=5 for n=40

Sn and Tn are convergent (to the same limit), but in one case, what you're looking for is n=10. In the other n=40. I can't think of a possible general "formula" or strategy.

I must admit I'm a little confused as to what you're asking.

 

[johnqwertyful]

Are there any constraints to the sequence? Are you looking how to do this in general, or just for the sequence you're working on? I've reread your post and don't quite understand what you're asking.

 

[Nelphine]

You actually seem to have it spot on - I'm really just wildly flailing and hoping someone has thought of something I haven't, because otherwise this particular theory of mine will probably not be fruitful. I'm really trying to figure out how to find properties of a sequence before it gets close to its limit, without knowing that much about the particular details of the sequence. Since I only ever took 2 courses that included such material, I was hoping that I just didn't have enough exposure and that someone might know something that could help.

And while I could give you some example sequences (such as the one which has the limit of 0.4), since I'm dealing an infinite number of sequences which are all slightly different, I don't think that would actually help much.