Speaker A
Hello everyone, welcome back to Math with Allison. Today we're working on our sequences and series, so specifically we're going to be talking about the limit of a sequence. So let's go and dive into it. Let's take a look at
Learn about limits of sequences, convergence, divergence, and examples with graphs in this detailed Calculus 2 lesson by Math With Allison.
Key Takeaways
- A sequence converges if it approaches a unique finite limit as n approaches infinity.
- Sequences that grow without bound or oscillate without settling diverge and have no limit.
- Graphing sequences as discrete points helps visualize their behavior and limits.
- Explicit formulas can be used to rigorously prove convergence or divergence.
- Oscillating sequences can still converge if their terms approach zero.
What the video covers
- Introduction to limits of sequences with examples of sequences a_n and b_n.
- Explanation of divergence to infinity and convergence to zero using explicit formulas.
- Definition of convergence and divergence with unique limits and no limits.
- Types of divergent sequences including those diverging to positive infinity, negative infinity, and oscillating sequences.
- Graphical visualization of sequences to illustrate convergence and divergence.
- Examples of convergent sequences including constant sequences and oscillating sequences converging to zero.
- Step-by-step calculation of limits for given sequences using formulas.
- Use of logic and formal proofs to determine limits of sequences.
- Discussion on bounded sequences and their convergence properties.
- Encouragement to practice with additional problems and explore related topics.
Chapters
- 00:00Introduction to Limits of Sequences
- 00:43Divergence to Infinity Explained
- 01:32Convergence to Zero and Definitions
- 02:22Types of Divergent Sequences
- 03:09Graphing Divergent Sequences
- 05:01Examples of Convergent Sequences
- 05:52Graphing Convergent Sequences
- 07:13Oscillating Sequences and Convergence
- 07:55Working Through Limit Problems
- 13:06Summary and Further Practice
Full Transcript — Download SRT & Markdown
Speaker A
these two different sequences. So first we have a sub n, which is equal to 1, 2, 4, 8, and so on and so forth. So notice what we're doing right here is we're always going to be multiplying by two. Now our b sub n
Speaker A
is equal to 1, 1/2, 1/4, 1/8, and so what's happening right here? We're always going to be dividing by two, right? So the idea of the limit of a sequence is what you think the sequence is going to approach.
Speaker A
So if we look at our first sequence a sub n, notice that it's getting bigger and bigger. So let's go to talk about that. a sub n is being multiplied by two each time and will only get larger. So here we
Speaker A
say the limit as n approaches infinity of our sequence is going to be infinity. So in this case, a sub n diverges to infinity. It means there's never a number that it's going to approach; it's only forever going to be getting bigger and
Speaker A
bigger. Now that's going to be different from b sub n, right? We have that b sub n is being divided by two each time and is only getting smaller and smaller. So we can find an explicit formula for it, right? I wrote it as one over 2 to the
Speaker A
power of n for n is equal to 0, 1, 2, and 3, and we can go ahead and take the limit of that to see what's happening. You could also just use logic. You can see that this is like 1 over 32, is getting really
Speaker A
really close to zero. But also if we took the limit of b sub n, we'd get 1 over 2 to the power of n, and as n gets really, really big, 1 divided by a big number goes to zero. So here we have proof that the
Speaker A
limit goes to zero. Since it goes to an actual number, we say that b sub n converges to zero. So let's go ahead and talk about those official definitions. We have if the terms of a sequence a sub n approach a unique number l, so in our
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previous case it approaches zero, and zero is a unique number. So here we have the limit as n approaches infinity of our sequence is equal to l. Then we say the limit of a sub n as n approaches
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infinity exists, and the sequence converges to l. It goes to a number. Now if the terms of the sequence do not approach a single number as n gets bigger and bigger, we say that the sequence has no limit, and the sequence
Speaker A
diverges. So in these two cases, we would say that a sub n has no limit. There's nothing to stop it, right? There's no number there that's going to be the barrier for it. It's just going to continue doubling; it's going to get
Speaker A
bigger and bigger. So here that's when we would say it diverges. So let's talk about the types of sequences that diverge. This is very general. This is not all of them, right? But there's some good examples in here. So this is our previous
Speaker A
example. We have that the limit as n approaches infinity of our sequence is going to go to infinity. It's going to continue doubling and only get bigger and bigger. So let's see what that looks like on a graph. I'm going to say a sub 0
Speaker A
is going to be our starting value, which is one. a sub 1 is going to be equal to 2, a sub 2 is equal to 4, a sub 3 then we go to eight, and then at four we go all the
Speaker A
way to 16. So notice it's very much like our function that goes something like this, and that function is only going to increase. But for sequences, it's just going to be the little dots, but you can see the pattern of it. So we say that the
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limit as n approaches infinity of our sequence is equal to infinity, and we say that a sub n diverges to infinity. Another way that you could write that is instead of saying it equals infinity, the limit does not exist,
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right? There's no number that's going to be the barrier. So here let's see another example. We have now that all of these are negative values, so just by using logic you can tell the limit is going to go down to negative infinity, right? But
Speaker A
let's go ahead and graph those out to see what they look like. So first we have -1, then we go down to -2, -4, 8, and then -6. So if you think about it like a function, it's going to be doing
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something like that, and as a sequence it's going to be those little dots. So we say that the limit does not exist, and we also say that b sub n diverges, but this time it diverges to negative infinity, right? So it can diverge to positive
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infinity. It can diverge to negative infinity. Let's see another example. What if it was alternating in sign? So 1, -2, 4, 8. Notice that this is oscillating between positive and negative values. This sequence will diverge, right? Because it approaches both
Speaker A
positive and negative infinity. So we would say that the limit as n approaches infinity of our sequence does not exist. And let's go ahead and graph those out. So first we have positive 1, then we go down to -2, then we go up to
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positive 4, 8, positive 16. So notice on either side we have something that's doing something like that, and another one that's doing something like that. Of course, they're not functions, so they're just going to be the dots, but that's
Speaker A
just to see the pattern. So another way that it can diverge is through oscillation. So now that we talked about types of sequences that diverge, let's talk about types of sequences that converge. So here we have the sequence
Speaker A
that we had in our first example. The conjecture is that a sub n is going to converge to zero, and that's just me using logic. I'm not using anything fancy here. So if we wrote it as an explicit formula, we get 1 over 2 to the power of n, and
Speaker A
when we take the limit as n approaches infinity, that's going to get really, really big in the denominator, and that's going to go to zero. So that would be more of an official proof. So here we say the limit exists, and a sub n
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converges, right, to zero. So let's go and see what that looks like graphed out. First we start at one, which is going to be all the way up here. We go down to 1/2, 1/4, 1/8, and 1/16. So if you think about it,
Speaker A
just looking for the pattern, we can see that we're getting really, really close to the x-axis, and so that's why it's converging. Let's see another type. This one is just where it's a repeated sequence: 3, 3, 3, 3. We say that the
Speaker A
limit is going to exist, and it's going to converge to three. So the limit is equal to three. So if you didn't like writing "exists" right here, you can say the limit exists; it is equal to three. And let's go ahead and see what that
Speaker A
looks like. It's actually just going to go ahead and be a straight line. It's like saying x is equal to three, right? It's always going to be three. Therefore, it has to converge to three. Let's try another one. So here we have 1, negative
Speaker A
1, 1/2, 1/4, negative 1/8. Notice that it's oscillating, but it can still converge even if it's oscillating. So the conjecture, just by using logic, is both those numbers, even though they're positive and negative, they're both getting very, very small, and on either
Speaker A
side they're getting closer and closer to zero. So my conjecture is that it converges to zero. If you wanted to use more of an official proof, you could write out the explicit formula. So here we have 1 to the n times 1 over 2 to the n. So if
Speaker A
we take the actual limit here, we have that 2 to the n gets so, so big that the whole thing goes to zero, whether it's positive or negative. So here we would say that this equals zero. So our limit
Speaker A
exists, right, and c sub n converges to zero. So let's see what this looks like graphed out. So first, first we start off at 1, then we go down to -1/2, positive 1/4, -1/8, positive 1/16. So on either side of
Speaker A
this we're just getting super, super close to zero. So it converges to zero, but that's what it would look like as a sequence. So let's do some real problems here. We have an explicit formula: n over n squared plus 1. We're going to write out
Speaker A
the first four terms of the sequence first so we can see a nice pattern. So let's do a sub 1. This is going to be 1 over 1 plus 1, which is going to be 1/2. a sub 2, I'm going to plug in two
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everywhere I see an n. That's going to be 2 over 4 plus 1, which is 2/5. a sub 3, that's going to be 3 over 9 plus 1, that's going to be 3/10. a sub 4, that's going to make it a positive one in the numerator, 16 plus 1,
Speaker A
that's going
Speaker A
we're going to talk about part two which is does the sequence converge or diverge and you need to explain your answer so first make a conjecture use some logic look at the sequence and see what you think it's
Speaker A
approaching I think this sequence and I'm I don't I guess I didn't name it so I'm just going to say this sequence I'm going to say this converges to zero because notice how it gets really really little even though it's oscillating
Speaker A
they're both getting very very small positive and negative SI so let's go ahead and actually prove this we can take the limit as n approaches Infinity of our sequence1 to n n^2 + 1 if you wanted to be really really specific you
Speaker A
could even use a squeeze theorem which is everyone's favorite so since this is oscillating between positive and negative we have that the negative one is always going to be less than or equal to 1 to the N which is our actual secant
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which is always less than or equal to the positive version now let's take the limit as n approaches Infinity of all three of these first we're going to do the left limit1 / n^2 + 1 when n gets really
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really big the denominator gets big that whole thing is going to go to zero even though it's negative so that with our actual sequence and then on the right n is going to Infinity 1 divided Infinity is going to go to zero so we have that this
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is bounded between 0o and zero which means our limit actually has to equal zero and so there's more proof that our sequence converges to zero of course it depends on how your teachers asking it so if they wanted you to show proof you
Speaker A
could do the limit if they just wanted you to explain you could say it's getting really really small like I the converges is zero here we have a recurrence relation given to us so we have a n + 1 = -2 a subn and we have our
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first term is going to be equal to one and we're going to write out the first four terms of the sequence so they already gave us a sub one so then a sub 2 is going to be equal to -2 * a sub 1
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which is equal to 1 a sub3 is going to be equal -2 * our previous term which is going to be equal to four a sub3 is -2 times our previous term which is8 so if I wrote this out as a sequence we have 1
Speaker A
-2 48 so again we're going to make a conjecture what do you think is happening well converge or diverge I think a subn will diverge because whether it's positive or negative that number remember in magnitude is getting really really big so in
Speaker A
magnitude a subn is continually doubling in size and will grow infinite another way you could talk about it is that it's approaching both positive and negative infinity and there's multiple other ways that you can do it of course but it's
Speaker A
going to be like how it's asked of you so yeah we're going to try one more here we have an explicit formula so a subn is equal to 4 n Cub / n + 1 we're going to enumerate and graph the terms of a subn
Speaker A
so that's another way that the question can be asked enumerate just means write out some of the terms so we're starting at one and usually the first four or five terms are pretty good to show you what's going on so we get 4 * 1 cubed 1
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+ 1 that's going to be 4id 2 which is equal to 2 a sub 2 is going to be 4 * 2 cubed which actually I'm just going to rewrite that as 8 4 * 8 / 2 Cub + 1
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that's going to be equal to 32 / by 9 a sub3 is going to be equal to that's going to be 4 * 27 over 27 + 1 which that is oh math you know it's never a bad thing to whip out the old fashion
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that's going to be 108 ided by 28 a sub4 is going to be equal to oh gosh okay four * 64 ided by 64 + 1 that is I know that one 256 divided 65 so I converted those
Speaker A
all to decimals we got two 3.5 repeating 3. 857 3.9 38 so notice what those are getting really close to let's go ahead and graph those out so our first term is equal to two our second term goes to
Speaker A
three and a half then we just get really really close to four right so if I just look at the pattern I can see something like this and let's go to make a conjecture on what it approaches I
Speaker A
believe a subn is going to converge to four and of course we're going to do the math to prove it we already have the explicit formula so we can take the limit as n approaches Infinity n cubed 4
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n Cub Over N Cub + 1 so when we're working with limits at Infinity it doesn't matter if we add one to Infinity right it's not going to make a difference so I can rewrite that as this so here those n cubed cancel out and
Speaker A
what we're left with is four and the limit of four is just going to be equal to four and so here we have proof that this converges to four so that's all I have for us in this video today if you
Speaker A
enjoyed it I have many more like it so make sure to check out my playlist or link down below otherwise please give this video a thumbs up and comment other problems or topics you'd like to see done thanks for
Speaker A
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Topics:Calculus 2SequencesSeriesLimits of sequencesConvergenceDivergenceMathematicsMath With AllisonInfinite limitsOscillating sequences
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