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# nth term divergence test

AP.CALC:
LIM‑7 (EU)
,
LIM‑7.A (LO)
,
LIM‑7.A.5 (EK)

## Video transcript

what we're going to do now is start to explore a series of tests to determine whether a series will converge or diverge and the first one I'm going to go through right now is perhaps the most basic and we'll hopefully see the most intuitive and this is the divergence test and the divergence test won't tell us if a series will converge but it can tell us if something will definitely diverge so first let me write it in kind of mathy notation and then we'll look at an actual concrete example of it so the divergence test tells us that if if the limit as n approaches infinity of a sub n does not equal zero then then the infinite series going from N equals 1 to infinity of a sub n will diverge will diverge and we've already gone through what it means the divers that this sum is either going to go unbounded to positive infinity or unbounded to negative infinity or it'll just oscillate between values it'll never really approach a given sum or given value so that's what the divergence test tells us and you're probably thinking ok all right I can kind of get what this says but where is this actually useful and to see where it can be useful let's look at a candidate series and see if we can figure out if it's going to diverge so let's say I had the series so the sum from N equals 1 to infinity and in general it doesn't always have to be N equals 1 it could be N equals 5 it could be N equals 0 the key is is that this is an infinite series that we're talking about so that's why we care about the limit as n approaches infinity and so let's say we have the sum of for N squared minus n to the 3rd over 7 over 7 minus 3n to the third are so given what we know about the divergence test is this series going to converge or diverge well let's just look at this this let's look at what take what we're taking the sums of so this is essentially or this is our a sub n if we're trying to match to the the definition or I guess the explanation of that divergence test so let's think about what the limit the limit as n approaches infinity of for N squared minus n to the third over seven minus three and to the third is and I encourage you to pause the video and think about that well there's a couple of ways to think about it one way say look as n goes to infinity the highest degree terms of the numerator in the denominator are the ones that matter and so this is going to approach negative n to the 3rd over negative 3 and to the 3rd which will just be which would approach negative 1 over negative 3 which would be which would really positive 1/3 or we could say if we want to do a little bit more or do a little bit more systematically limit as n approaches infinity we can divide what the numerator and the denominator by n to the third so if we divide the numerator by n to the third this first choice it's going to be 4 over N minus 1 over 7 over n to the third minus 3 and here it becomes clear the limit as n approaches infinity that's going to go to 0 that's going to go to 0 and so you're going to be left with negative 1 over negative 3 which is equal to 1/3 so notice the limit of a sub n as n approaches infinity in this case is not equal to 0 therefore this sum will this infinite series will diverge diverge now let's take for a second why this makes a ton of sense well the only way that you're going to converge you're going to settle you what we're doing you're doing an infinite sum you're taking an infinite sum of things so the only reasonable way that something might be able to converge to a finite values if every extra term you're adding is getting smaller and smaller and smaller is approaching 0 if as n approaches infinity you you go unbounded or if it's even equal to one-third this means that for very large ends you just keep adding things that are getting closer and closer to one third well if you add an infinite number of one thirds together you're going to go to infinity you're going to be unbounded you are going to diverge so that's all that's telling us look in order for something to converge after doing an infinite sum of them at some point these things are going to get really really really close to zero if at some point they're not getting close to zero there's no way it's going to converge that thing will diverge so hopefully that makes a little sense and that also gives us another insight on what that first test can to do the divergence test can be used to show that something will diverge but but if something I guess you could say passes the divergence test it doesn't or if I guess it or I guess fails the diverge this 2's or if the if this isn't true it doesn't mean that the thing is going to converge and so let me give you an example of that so this right over here from N equals 1 to infinity of 1 over N and this is actually the harmonic series right over here this if we look at if we try to apply the divergence test we would say okay well what's the limit as n approaches infinity of 1 over n well way that is 0 as n approaches infinity this is equal to 0 and so we could say well it's it's kind of failing the divergence test so we're not just by using the divergence test we can't prove that this thing is going to diverge but that doesn't mean that it doesn't diverge it actually turns out and we we prove this in several videos it actually turns out that this thing does diverge this thing does does diverge it's just that the divergence test isn't enough it's not as enough of a tool to let us know for sure that this diverged you will see the comparison test and the integral test can either be used to prove that this in fact does diverge and so you can't you can definitely not say that if something if this does not apply for something if you're taking the limit as n approaches infinity and it does go to 0 that's the thing still might diverge it doesn't necessarily mean that it converges now there are things that do converge that where they do approach 0 so for example if I take the sum from N equals one to infinity of one over N squared if I try to apply apply the divergence test I have the limit as n approaches infinity of 1 over N squared well this does equal zero this gets really really large this thing is going to approach zero and so once again this is I guess failing the test for divergence just with that alone you don't know for sure that this thing is going to converge it still might diverge the divergence test just might not have been enough now it does turn out and once again we prove this in future videos that this thing does converge this thing does converge but not because not because it I guess you could say fails the divergence test we have to use a different test to show that this does in fact converge just as we have to use a different test to show that this does in fact diverge where the divergence test is useful is for the things that actually pass the divergence test when you actually find that the limit as n approaches infinity of a sub n does not equal zero like this case right over here in this case that a vergence test helps us because we helps us make the conclusion that this series definitely definitely die the series definitely diverges