yego.me
💡 Stop wasting time. Read Youtube instead of watch. Download Chrome Extension

Worked example: limit comparison test | Series | AP Calculus BC | Khan Academy


3m read
·Nov 11, 2024

So we're given a series here and they say what series should we use in the limit comparison test. Let me underline that: the limit comparison test in order to determine whether ( S ) converges.

So let's just remind ourselves about the limit comparison test. If we say, if we say that we have two series, and I'll just use this notation ( a_n ) and then another series ( b_n ), and we know that ( a_n ) and ( b_n ) are greater than or equal to zero for all ( n ). If we know this, then if the limit as ( n ) approaches infinity of ( \frac{a_n}{b_n} ) is equal to some positive constant (so ( 0 < c < \infty )), then either both converge or both diverge.

It really makes a lot of sense because it's saying, look, as we get into our really large values of ( n ), as we go really far out there in terms of the terms, if our behavior starts to look the same, then it makes sense that both these series would converge or diverge. We have an introductory video on this in another video.

So let's think about, what if we say that this is our ( a_n )? If we say that this is ( a_n ) right over here, what is a series that we can really compare to that seems to have the same behavior as ( n ) gets really large? Well, this one seems to get unbounded. This one doesn't look that similar; it has a ( 3^{n-1} ) in the denominator, but the numerator doesn't behave the same.

This one over here is interesting because we could write this. This is the same thing as ( \sum_{n=1}^{\infty} ) we could write this as ( \frac{2^n}{3^n} ), and these are very similar. The only difference between this and this is that in the denominator here (or in the denominator up here) we have a minus one, and down here, we don't have that minus one. So it makes sense, given that that's just a constant, that as ( n ) gets very large, these might behave the same.

So let's try it out. Let's find the limit. We also know that the ( a_n ) and ( b_n )--if we say that this right over here is ( b_n ), if we say that's ( b_n ), that this is going to be positive or this is going to be greater than or equal to zero for ( n = 1, 2, 3 ). So for any values, this is going to be greater than or equal to zero, and the same thing right over here; it's going to be greater than or equal to zero for all the ( n ) that we care about.

So we meet these first constraints, and so let's find the limit as ( n ) approaches infinity of ( a_n ), which is written in that red color: ( \frac{2^n}{3^{n-1}} ) over ( b_n ) over ( \frac{2^n}{3^n} ).

So let me actually do a little algebraic manipulation right over here. This is going to be the same thing as ( \frac{2^n}{3^{n-1}} \cdot \frac{3^n}{2^n} ). Divide the numerator and denominators by ( 2^n ); those cancel out. So this will give us ( \frac{3^n}{3^{n-1}} ).

Like we can divide the numerator and denominator by ( 3^n ), and that will give us ( \frac{1}{1 - \frac{1}{3^n}} ). So we could say this is the same thing as the limit as ( n ) approaches infinity of ( \frac{1}{1 - \frac{1}{3^n}} ).

Well, what's this going to be equal to? Well, as ( n ) approaches infinity, this thing ( \frac{1}{3^n} ) is going to go to zero. So this whole thing is just going to approach one. One is clearly between zero and infinity, so the destinies of these two series are tied. They either both converge or they both diverge, and so this is a good one to use the limit comparison test with.

And so let's think about it. Do they either both converge or do they both diverge? Well, this is a geometric series; our common ratio here is less than one, so this is going to converge. This is going to converge, and because this one converges, by the limit comparison test, our original series ( S ) converges.

And we are done.

More Articles

View All
Miranda v. Arizona | Civil liberties and civil rights | US government and civics | Khan Academy
[Kim] You have the right to remain silent. Anything you say can and will be used against you in a court of law. We’ve become familiar with the Miranda Warnings given to suspects in police custody through movies and TV shows, but who was Miranda and what d…
Knowledge Makes the Existence of Resources Infinite
Knowledge is the thing that makes the existence of resources infinite. The creation of knowledge is unbounded. We’re just going to keep on creating more knowledge and thereby learning about more and different resources. There’s this wonderful parable of …
Chandragupta, Ashoka and the Maurya Empire | World History | Khan Academy
We’re now going to talk about the Moria Empire, which is not just one of the greatest empires in Indian history, and really the first truly great Empire. It’s also one of the great empires of world history. Just for a little bit of context, we can see whe…
Best PHOTOBOMBS: IMG! episode 12
The Cheez Whiz bird of Oz and a brand new Wii controller. It’s episode 12 of IMG Woody and Buzz Lightyear all grown up and Dora the Explorer all grown up, or as Jessica Alba here is Star Wars as a classic PC adventure game. My favorite is the 12 pixel sla…
15 Philosophies That Will Change Your Life
A single sentence could change your life. These philosophies are meant to shake you out of complacency. They’re meant to bring you back down to earth to make you aware of your presence in the world. When it hits home, it’ll give you the inspiration to get…
Teaching a Fixated Dog to Focus | Cesar Millan: Better Human Better Dog
For me, it’s easier to rehabilitate an aggressive dog than a fixated dog. While working with fixated and overexcited kelpie Shadow, Caesar discovers the dog has forgotten how to behave like a dog. “That’s my girl! Let’s go swimming!” To prevent aggressio…