Main content

## Integral Calculus

### Unit 5: Lesson 18

Proof videos- Formal definition for limit of a sequence
- Proving a sequence converges using the formal definition
- Finite geometric series formula
- Infinite geometric series formula intuition
- Proof of infinite geometric series as a limit
- Proof of p-series convergence criteria

© 2023 Khan AcademyTerms of usePrivacy PolicyCookie Notice

# 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.

## Want to join the conversation?

- What does epsilon stand for in this case? a random value that is greater or less than L?(63 votes)
- 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.(192 votes)

- 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?(6 votes)
- 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.(15 votes)

- 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.(14 votes)

- 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!!(4 votes)
- 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.(3 votes)

- What does M mean on the n axis?(4 votes)
- That is the point where if n > M the error bound of the output will be within epsilon of L.(4 votes)

- If epsilon is the size of the interval, shouldn't L be shifted by +/- epsilon/2 so that the total gap is epsilon?(2 votes)
- 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.(5 votes)

- What exactly are epsilon and M ??(2 votes)
- 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*with in some arbitrarily small distance.**and**stay

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

If that was not clear for you, try this http://www.sosmath.com/calculus/sequence/limit/limit.html(5 votes)

- 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?(2 votes)
- 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.(2 votes)

- 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?(2 votes)- 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.(3 votes)

- i dint undestand y E+L AND E-L IS THE LIMIT(1 vote)
- 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`

.(3 votes)

## Video transcript

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.