Definition of limit of a sequence and sequence convergence


Definition of limit of a sequence and sequence convergence

Discussion and questions for this video
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.
Sal is trying to prove that the sequence converges to the value L. Epsilon is like a bound on the curve being plotted. Consider the plot, at any point, the value of x can be some small value greater than L or a small value less than L.

(a_n - L) will give you this difference. Since we are interested only in the magnitude, we use the absolute value of (a_n - L) => |a_n - L|

He says that, if we are able to prove that the difference is less than E(epsilon, a arbitrarily chosen small value) for a value of x greater than M, the sequence converges.

This allows one to chose any small value for E(epsilon). If E(epsilon) is zero, it means that the difference is zero. This could happen only when the sequence has converged to L. Hence, if we are able to prove, for any value of E, the diff is less than E, the sequence would converge
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.
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?
The way I've been lead to understand it, is that there are three main branches to Calculus: Differentiation, Integration, and Infinite Series. It seems they are usually taught in this order, and limits are a vital foundation for how and why derivatives work. Then they drift out of the spotlight as we learn integration, and come back in with a vengeance for infinite series. Part of the reason for this is probably that integration plays a part in one of the tests you can use to tell if a series converges or not.
I was a bit disappointed about how little Sal has done on infinite series in the calculus section, and was surprised to find anything about it in precalc.
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.
The limit is `L`, `L-ε` and `L+ε`, are ever decreasing margins that the sequence must be within as `n` grows, so when `n` is very big, `ε` is very small, and the sequence has converged to the value `L`.
Sir, whats the difference between 'Limit of a Sequence' and 'Limit Point of a Sequence'. Do they mean the same. Whats actually the difference between the two,Sir..
So for a sequence to be convergent, does every single value of n have to satisfy this definition, or at least one?
After knowing the definition of the limit of a sequence and sequence converges? It is possible to provide a definition of the limit of sequence diverges?. as the definition for it seems vague and very different from the convergence limit. Millions of thx.
How do you find the relationship between two sequences? Or even their limits?
He is referring to the numbering of items in the sequence. If we begin numbering at 1, then the first item in a sequence is x sub 1 and the second is x sub 2 and so on. These subscripts may be called the indices (plural of index).

Sal points out here that we can treat a sequence as a function of its index. Recall that a function is a rule that produces a unique output for each input. If we use the index of a function as the input we get a function: for input 1 the output is x sub 1, for input 2 the output is x sub 2, and so on. When we treat a sequence as a function of its index, we can graph it as Sal does in this video, and obtain a clearer picture of how the sequence behaves.
how i can determine wheathe rthe series converges r diverges ?
Hey why dont you watch the video on convergent and divergent sequences.. Well if the members of the sequence seem to be approaching a certain value, it is convergent and if the members are going farther then they are divergent..
For example: convergent sequence - {6, -5, 3, -1, 1.......}. (Converging to 0 )
Divergent sequence- {2, -4, 5, -6, 9.....}.
not so well explained!does anyone know where I could get a better graphical representation?I can't make an image
This is an arbitrary assumption for calculations
So essentially, if any n is closer to L than aM (subscript), the series converges
It is not that there exists some n>m, but for all n>m. If L is the limit, then for any tolerance, you can find a term of the sequence such that beyond that term in the sequence you will be within the stated tolerance of L.