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# Formal definition for limit of a sequence

A sequence is "converging" if its terms approach a specific value at infinity. This video is a more formal definition of what it means for a sequence to converge. Created by Sal Khan.

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• What does epsilon stand for in this case? a random value that is greater or less than L? •   This gets kinda long, but stick with me.
Epsilon (ε, lowercase) always stands for an arbitrarily small number, usually < 1. It has a counterpart, delta (δ, lowercase) which is associated with the x-axis. Together they are used to strictly define what a limit is, among other things. Another place you may use epsilon is in computer programming, and I think a programming example serves well here. Floating point variables have far more decimal places than we will ever care to look at, and often more than we care to keep track of for our calculations. If we need to know if a number equals zero after some calculation, there's a very good chance it never will. The reason is that a result may come out to something like 0.0000000000000000000000000001258453359, which is certainly not EQUAL to 0. However, there are few cases where we would care about this much difference from 0, and wouldn't just round it off and call it 0. This is where the epsilon comes in. Say we decide that being within 4 decimal places is close enough. Our epsilon is 0.00001, and here, our L is 0. If we get a result (call it 'a') close to 0 like a = 0.000032, we subtract L from it, take the absolute value and compare it to epsilon. So:
| 0.0000032 - 0 | = 0.0000032 < epsilon
In this case we decide to call it 0 and move on with our calculations.
Another example:
a = 0.00013
| 0.00013 - 0 | > epsilon
so we say it's not equal to zero.
This works for other values as well. Say our L is 2 (this might be the L in the video). We still want to know when our a is close enough to L to just call it L. Our epsilon stays the same (0.00001). Say a = 2.000000145
| 2.000000145 - 2 | = 0.000000145 < epsilon
So here the test shows that a is close enough to L as makes no difference.
Try this one:
a = 1.99934
| 1.99934 - 2 | = 0.00066 > epsilon
Here we're close, but not close enough.

The epsilon you choose can be any number. Usually it's less than one, but if we estimate that the epsilon in the video was 1, we could just as easily have chosen 1.5 and included the first couple of points in the epsilon bounds. The point here is that the epsilon bounds don't have to include all the points in the series, just the points greater than M, which we choose arbitrarily. If M is 0, our epsilon bounds have to be far apart, but all the a's will fall inside it (for this example). If M is 20, our epsilon bounds can be very small, and will include all the points after a_20, way off the graph to the right. As long as any a_n where n > M falls within the epsilon bounds, the series will converge.

Sal could do (has done?) a whole video explaining epsilon stuff. I "learned" this in Calc I, and it's only just starting to make good sense as I try to explain it :)
Hope it helped.
• This seems very similar to the Epsilon-Delta definition of limits discussed earlier in the differential calculus playlist, but it seems that Sal used "M" here instead of "delta". Is there any particular reason that we use "M" instead of "delta" when dealing with limits for converging sequences? • With sequences we are moving discretely over the integers. Therefore M is not a range, as delta is, M is the index of the sequence such that if index value, n is greater than some M then for all n > M, that is, a_M, a_M+1 as n -> to infinity, then ALL the terms of the sequence will be within epsilon of the actual limit, L.
• would epsilon in this case be a number greater than 0 and smaller to all other positive real numbers? Would that be a valid definition?
(1 vote) • Not quite. The idea is not that epsilon is some specific infinitesimal number smaller than all other numbers, but rather that epsilon could be any number > 0, and that no matter how small we choose epsilon to be, there is another number, M, such that once you go past M, a_n is always within the distance epsilon of L. So usually the idea is that M in some way depends on epsilon, in the sense that the smaller epsilon is, the bigger M will be, in order to get to the point where the sequence gets close enough to L and never gets further than epsilon, no matter how far you go. So it's true that we usually think of epsilon as being "very small", but it's actually important to realise that it is also definitely some finite number, which corresponds to some finite M. If epsilon were smaller than any real number, then (in most cases), M would have to be infinity.
• Sal uses dorm pretty advanced 'jargon' like episilon and also uses some unknown signs. Like what, do the 2 lines bracketing a sub n minus L mean?? Has he explained this stuff in a previous video?? Sorry for being so vague!! • The two lines used as brackets are used to denote Absolute Value. Sal does cover this in the Arithmetic section.

As for epsilon, I am not sure. I was reading these questions to find the answer myself. But I think it is used to prove something is true or false. You pick a random, very small number and then try to pick an M and n that makes the equation work.

I am sure this is very generalized and may be wrong. I would like to know the answer myself.
• • If epsilon is the size of the interval, shouldn't L be shifted by +/- epsilon/2 so that the total gap is epsilon? • If I asked you is 9.0001 within 1 of 10 what would you say?
Presumably you would do the subtraction 10 - 9.0001 = 0.9999 < 1 and say yes.
Are both 9.0001 and 10.9999 within 1 of 10?
They are, yet the distance between them is 1.9998 which is greater than 1.

It is the same thing with epsilon and the limit value.
Suppose the limit, L, is 10 and epsilon is 1. and we have n greater than some M for some sequence with terms a_n, then if 9.0001 < a_n < 10.9999, that means |a_n - L| < epsilon for our M>n, thus the epsilon definition of the limit of the the sequence is satisfied and the sequence has a limit.
This way of defining the limit allows us to be within epsilon of either side of L.
• • What we want is have a clear understanding of what it means to say a sequence is converging. How do we know? Well, we can say the sequence has a limit if we can show that past a certain point in the sequence, the distance between the terms of the sequence, a_n, and the limit, L, will be and stay with in some arbitrarily small distance.
Epsilon, ε, is this arbitrarily small distance.
M is the index of the sequence for which, once we are past it, all terms of the sequence are within ε of L.

Recall that we can define the distance, d, between two points as |a-b|=d.
Recall that a sequence is an ordered list of indexed elements, eg S=a_1, a_2, a_3,...a_n, and on to infinity.
What we have in this situation is that once the index of the sequence is greater than some index value, let's call it M, the distance between nth element of the sequence, a_n, and the Limit, L, is less than epsilon, ε. We write that as |a_n - L| < ε, where n>M.

Now here is the important part. If you can say this for any size of ε you care to choose, and |a_n - L| < ε holds, then L is the limit of the sequence.

If that was not clear for you, try this http://www.sosmath.com/calculus/sequence/limit/limit.html
• For the example sequence in the video, it's easy to imagine that the sequence converges to L. But how do we know that the sequence doesn't do something weird at some very large n, such as make a large jump or drop. This value of the sequence would not be within epsilon of L. In terms of the video, my question is, how do we know that there are no jumps in the sequence past M. It may be obvious for some sequences that there are no large jumps/drops for any n past the M value but could there be some sequences for which it's not so clear? Can the limit be proven? • The idea is that if you can find a M such that there is no jump after M that is larger than epsilon then the sequence converges. If you can't find an M like that then the sequence diverges.

As for proving that a specific sequence converges (finding an M such that all terms after M are within epsilon), that will depend on the sequence itself. It might be tricky to do if the sequence is nutty. I think someone mentioned in one of the other answers that there are techniques that use integration that can help. I'm sure there are a lot of other techniques as well.
• In a math book that I have, the author describes the basics of Calculus in terms of limits of sequences and not as limits of functions (like the way Sal does in his Calculus playlist). Is it because describing limits in terms of sequences is "more rigorous" or "more general" than the other method?

I get that both methods are conveying the same ideas and that, technically, sequences are functions, too. However, I'd like to know if one method has certain benefits over the other (for instance, if one method is preferred by mathematicians, etc.).
Do you guys know what I'm talking about? • 