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.
That is the point where if n > M the error bound of the output will be within epsilon of 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?
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.
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.
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.
The Cauchy test differs ever so slightly, but in a very important way, from what is presented in the video.
In the video we have: if given ε>0, then there exists an M such that if n>M then | An - L | < ε. This test can only be used if you know what the value of the limit L actually is.
With Cauchy, you don't need to know what the limit is since it compares the distance between terms as n increases:
If given ε>0, then there exists an M such that if n,m > N then | An - Am | < ε.
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..
M is a particular value of n, where n is the index of a sequence and n>=0.
Therefore M must be greater than zero too.
What does "M" have to do with anything? The equivalent of delta? If you picked M>n , what would happen?
Yes, the idea is similar to the epsilon delta concept for defining limits.

What we want is for the terms of the sequence to become arbitrarily close to the limit L.
Sequences are indexed by integers, eg a_1, a_2, a_n. The M is just an index such that the value of every term in the sequence after it (M+1, M+2, M+n etc.) stays within ϵ of the limit L, or more formally IF n > M THEN | a_n - L | < ϵ. Since we can make ϵ arbitrarily small, we can say that the sequence does indeed have a limit at 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
Epsilon does not assume one specific value; when defining convergence, we want the assertion to hold for _every_ real number epsilon that is strictly greater than zero. In a sense, epsilon may be taken as small as you want, as longs as it is greater than zero.
Are we just trying to prove that the difference between {a_n - L} will be very small, when a sequence converges.?
You got the logic backwards. Informally speaking, it should be "if we can make the quantity `|a(n) - L|` small, then `{a(n)}` converges". The way you formulate it implies that the notion of convergence is already defined, and when a sequence converges, then we can make this difference small. Observe that it does not make sense to speak about convergence without first defining it this way.
What is the relation between M and epsilon? What if we pick M as 0 while epsilon stays the same as in the video? Then A1-L would be larger than epsilon, right? Sorry if it's a silly question, I think I didn't get the concept. Thanks in advance.
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.
The index is the counting number n (or k, or i or whatever). What he says is that we often view a sequence as a function of the indices. In other words, for each value of n, there is a specified value of the sequence based on the definition in terms of n.
If the index is n, and the sequence is defined as starting at n= 0 or n = 1, then for every value of n, we can generate a new term of the sequence. If the sequence is (-1)ⁿ⁺¹ ∙ 1/n²
when the index = 2, the term is -¼

Hope that helps
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
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.