Sequences, series, and function approximation
Sequence convergence and divergence
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Definition of limit of a sequence and sequence convergence
Discussion and questions for this video
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 what i want to do in this video is to provide ourselves with a rigorous definition
 of what it means to take the limit of a sequence as n approaches infinity
 and what we'll see is actually very similar to the definition of any
 function as a limit approaches infinity and this is because the sequences really can be just viewed
 as a function of their indices, so let's say
 let me draw an arbitrary sequence right over here
 so actually let me draw like this just to make it clear
 but the limit is approaching so let me draw a sequence
 let me draw a sequence that is jumping around
 little bit, so lets say when n=1, a(1) is there,
 when n = 2, a(2) is there, when n = 3, a(3) is over there
 when n=4, a(4) is over here, when n=5 a(5) is over here
 and it looks like is n is so this is 1 2 3 4 5
 so it looks like that as n gets bigger and bigger and bigger
 a(n) seems to be approaching, seems to be approaching some value
 it seems to be getting closer and closer, seems to be converging
 to some value L right over here. What we need to do is come up with a definition of
 what is it really mean to converge to L.
 So let's say for any, so we're gonna say that you converge to L for any,
 for any ε > 0, for any positive epsilon, you can,
 you can come up, you can get or you can, there is
 let me rewrite it this way, for any positive epsilon there is a positive,
 positive M, capital M, such that,
 such that if, if, lower case n is greater than capital M,
 then the distance between a(n) and our limit, this L right over here
 the distance between those two points is less than epsilon.
 If you can do this for any epsilon, for any epsilon,
 greater than 0, there is a positive M, such that if n is greater than M,
 the distance between a(n) and our limit is less than epsilon
 then we can say, then we can say
 that the limit of a(n) as n approaches infinity is equal to L and
 we can say that a(n) converges, converges, converges to L.
 So let's, let's, let's parse this, so here I was making the claim
 that a(n) is approaching this L right over here,
 I tried to draw it as a horizontal line.
 This definition of the of what it means to converge for sequence to converge
 says look for any epsilon greater than zero.
 So let me pick an epsilon greater than zero, so I am gonna go to L plus epsilon,
 actually let me do it right over here, L, so see this is L plus epsilon
 and let's say this is right here this is L minus epsilon.
 So let me draw those two bounds, right over here.
 And so I picked an epsilon here so for any,
 for any arbitrary epsilon, any arbitrary positive epsilon I pick, we can find a positive
 M, we can find a positive M, so that as long as,
 so let's say that is our M right over there.
 So that as long as our n is greater than our M, then our a(n)
 our a(n) is within epsilon of L, so being within the epsilon,
 being within epsilon of L is essentially being in this range.
 This right over here is just saying, look that the distance between a(n) and L is less than epsilon,
 so that would be any of these, anything that is in this, between L minus epsilon and L plus epsilon.
 The distance between that and our limit is going to be less than epsilon.
 And we see right over here, at least visually,
 if we pick M there and if you can take an n that's larger than that M,
 if you pick an N that's larger than M, if M is equal to 3, a(n) seems to be
 close enough. If M is 4, a(n) is even getting closer. It's within our epsilon.
 So if we can say, if we can say that it is true, for any epsilon that we pick, then we can say,
 we can say that the limit exists, that a(n) converges to L.
 In the next video we will use this definition to actually prove that a sequence converges.
Be specific, and indicate a time in the video:
At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?

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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 xaxis. 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.
Epsilon (ε, lowercase) always stands for an arbitrarily small number, usually < 1. It has a counterpart, delta (δ, lowercase) which is associated with the xaxis. 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.
Yes, but epsilon is more of a tolerance level. Because your series will never be equal to L, we assume that if the series is within the range of epsilon it is equal to L. Hopefully that helps.
I think so. Any chosen distance from the limit, measured vertically up or down.
What is M and epsilon refers to in this video?
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
(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.
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.
Epsilon is just a number, a randomly chosen small number. The "two lines bracketing the a sub n minus L" represents the distance between the limit L and the wavy function a sub n. As brian knox said, the two lines are absolute value lines. Since we want to know the distance between the function and the limit, we take the absolute value of a sub N minus L, since there's no such thing as a negative distance.
M is a number on the n axis to the right of which all the values of n are less than epsilon away from the limit. Remember epsilon is just a number that we have chosen on the axis where the Limit L is, above and below L. That's the way my simple brain understands it. Hope it helps.
M is a number on the n axis to the right of which all the values of n are less than epsilon away from the limit. Remember epsilon is just a number that we have chosen on the axis where the Limit L is, above and below L. That's the way my simple brain understands it. Hope it helps.
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?
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.
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.
I'd say no, maybe someone else my help clear this up.
But afaik a function is just the literal and general term used for the description of what a a superset of sequences really is. The word imo uses this kind of mapping ( injective, bijective, surjective ) because it is easier to categorize certain types of functions in contrast to other uses  e.g. in programming, which needs more complex sequences.
But afaik a function is just the literal and general term used for the description of what a a superset of sequences really is. The word imo uses this kind of mapping ( injective, bijective, surjective ) because it is easier to categorize certain types of functions in contrast to other uses  e.g. in programming, which needs more complex sequences.
What does M mean on the n axis?
That is the point where if n > M the error bound of the output will be within epsilon of L.
This seems very similar to the EpsilonDelta 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.
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.
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.
What does appselon means here ?
Sal used its symbol(€) there in the video .
Sal used its symbol(€) there in the video .
Epslon is the smallest positive value..
cauchy test theorem
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.
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.
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.
are tan(x) and cot(x) functions diverging functions?
If you mean _unbounded_ functions, then yes.
i dint undestand y E+L AND EL IS THE LIMIT
The limit is `L`, `Lε` 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`.
Lε and L+ε are the neighborhoods of L.We have chosen ε as it is the smallest value as such the value of ε has no effect on the value of 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..
why can't M be negative
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.
Therefore M must be greater than zero too.
What is the value of Epsilon?
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.
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.
At 0:30, what are the indices of a sequence?
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
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?
at 0:22, what does he mean by 'indices'?
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.
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.....}.
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
Why does epsilon need to be positive if it's a difference?
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.
because it calculate what you calcute i think mkb fpjb
the epsilon is the euro symbol in the keyboard?
The are similar but not the same. Notice that the euro symbol, €, has two cross marks. The Greek letter epsilon, ε, is more like a small version of the cursive English capital E: ℰ
How would do a sequence with a power? Or a factorial?
u need to find epsilon and M to find the limit
What does epsilon represent? Is it somehow related to the standard deviation? I am so confused!
epsilon is a real number , greater than 0 , which we pick depends how useful is for ourselves.
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