Choosing Which Convergence Test to Apply to 8 Series

Dr. Trefor Bazett explains how to choose the right convergence test for different series with 8 example series.

Key Takeaways

Choosing the right convergence test depends on the form and behavior of the series terms.
Start with simple tests like the divergence test before moving to more complex ones.
Geometric, integral, alternating, comparison, limit comparison, root, and ratio tests each have specific scenarios where they are most effective.
Integral test is often a last resort due to complexity.
Understanding the conditions for each test helps avoid misapplication.

What the video covers

The video addresses the challenge students face in selecting the appropriate convergence test for various series.
Dr. Bazett presents eight different series and matches each with the most suitable convergence test.
The geometric series test is applied first by identifying a fixed ratio in the series.
The integral test is discussed as a last resort, used when other simpler methods do not apply.
Alternating series test is applied to series with terms involving (-1)^n, with conditions on positivity, monotonicity, and limit zero.
The divergence test is recommended as a quick initial test to check if the sequence terms do not approach zero.
Comparison and limit comparison tests are used for series resembling p-series or geometric series, depending on inequalities and term behavior.
The root test is applied to series with terms raised to the nth power, focusing on the nth root limit.
The ratio test is introduced as a final method by process of elimination for series not covered by other tests.
The video concludes with a summary of all eight series and their corresponding convergence tests.

Chapters

How do I know when to use the geometric series test?

Use the geometric series test when the series has a fixed ratio between consecutive terms, often identifiable by terms raised to the power of n with a constant base ratio.

When should I apply the integral test to a series?

The integral test is best used when the series terms correspond to a function that is integrable over an interval, typically as a last resort when simpler tests do not apply.

What is the first test I should try when analyzing series convergence?

It is recommended to start with the divergence test by checking if the sequence terms approach zero; if not, the series diverges immediately.

Full Transcript — Download SRT & Markdown

Speaker A

one of the hardest things for students when they're first been introduced to series is how do I know which test to use for which kind of series there's so many different tests out there and indeed what I'm doing in this video is I'm giving you eight different series and we're doing them matching game to figure out what kind of test should I apply to which series and then all the way at the end of the video I'm going to step off the screen and I'm just gonna put up the full answers for each of

Speaker A

those and if you want to fast-forward and check on any particular one you're welcome to do that but I'm not gonna present through the answers I'm just gonna present through how I choose the method to apply so first up I want to go and look at the geometric series so there's all of these ones that the figure out which is gonna be the geometric series two metric series was how we began the course so the point is I want to find something with a fixed ratio and indeed if I look at this

Speaker A

second one that we have here but that second one has a sort of ratio in it it's a two to the power of n and a five to the power of n it it kind of looks like two fifths to the power of n the only problem here before I can directly apply the geometric series is that there's an N minus 1 and n plus 1 it doesn't quit lines so I can pull out two factors of 5 from that denominator turn that plus 1 into a minus 1 then it lines up exactly the geometric series I can

Speaker A

apply it R is 2/5 so it would converge so that was the geometric series we're already done with one of them now well what's next the next one we saw was actually the interval test and the integral test said that if you have a series you can look at the corresponding interval where the function is just the replacement of M with X and if that interval converges then so too does the original series now many of these series it doesn't make sense to use an interval

Speaker A

test on for example we can't really have a nice function representation of the factorial term minus 1 to the X is messy although there's some ways to deal with that in terms of cosines but it's just challenging for us some of the ones we have on this side with square roots into the power of ends they're just if you try to integrate the gonna be really hard but there is one that is screaming out to be integrated which is this one with this negative exponential down here in this scenario

Speaker A

if I consider the improper interval 1 up to infinity of X e to the minus x squared well for that I can do this it's going to be a you substitution it will take a little bit of work you could see my full solution at the end but I could do this it's gonna turn out to converge and so too does the original in truth while I introduced the integral test relatively early on when we were talking about series I usually leave it a bit as a last resort if I can apply a lot of

Speaker A

the other methods they're often simpler and I'll do the integral test at the end of nothing else seems to be working it appears like something I could integrate now carrying on the next one I want to talk about is alternating series test and the thing to note here is that there's actually two different series that I've put up on this list that have this minus one to the end so it may be one or both or neither of these so there's two series that are candidates for alternating series test el-amin

Speaker A

series this is perhaps the clearest you don't have a minus one to the power of n or n minus one then don't use the alternating series test now there was a condition on the alternating series test it has this minus one to the end but the other portion of it whatever was left over had to be positive decreasing and had limit zero now the top one of these has those conditions it's decreasing is positive and in the limit of 1 over root n is 0 so the alternating series test

Speaker A

applied to the top one and it converges but for the bottom one logarithm of n is positive but it's not decreasing and it doesn't have limit zero so actually it's only the top one that's going to be alternating series test this one we still have to figure out now what can we do here this one is a divergence test so how do i do diverges test diverges test is I don't look at the series I look at the sequence and if the sequence goes to something that's nonzero then it

Speaker A

diverges indeed if I look over here there's the plus and minus thing but the logarithm goes to infinity so as n goes large it goes really large positive really large negative even larger positive even negative it certainly doesn't convert to serum the sequence doesn't convert to zero and if the sequence doesn't converge to zero then the series diverges I actually like doing divergence test often first it's often the very first one I test because it's pretty easy to do doing the limit of the

Speaker A

sequence is a relatively straightforward calculation it's sort of repeat what we did not in calculus one so I'll often do a quick divergence test first before I go any further testing of the methods so I put this forth but if you want to do it first every time go nuts all right so that gets four of these and now I'm gonna move them over side a bit of space to work here I've got these other four to do the next one I want to talk about is comparison test and limit comparison

Speaker A

test we'll do a comparison test first now the whole point about the comparison test is that you have to have it compared to something you can do and there's really only two things that we know how to do very well there's geometric series and there's P series we know what geometric series converge do we know when P series converge now both the top and the bottom in this scenario have something that looks like a quotient of polynomials or with square roots but in the limit as n goes to

Speaker A

infinity the top and bottom look a lot like P series they look like some power of n so these are the types of ones one apply to comparison test two or the limit comparison test two particularly if you've got these nice fractions the middle two don't really work for a comparison test n factorial there's nothing easy to compare that to to make it look like a p-series if I look at this one that's to the power of n I don't see how I could make that look like a P series or a geometric series

Speaker A

but the top and the bottom one those I have a hope for now for this top one where I'm going to apply the comparison test it is a N on the top at an n cubed plus 1 on the bottom so what I'm going to say that this is less than what I'm going to compare it to is a BN that is just 1 over N squared think of this is actually main n on the top and an N cubed on the bottom that cancels to the 1 over N squared notice this is very dependent on the inequality here the fact that it's a

Speaker A

positive 1 in the bottom means the a n is in the VN and we can conclude that because the BN converges it's a P series with P equal to 2 that so too does the a.m. by my comparison test so that brings me to the limit comparison test now there's only three remaining I already suggested the bottom one it's gonna make sense for these comparison and limit comparison ones but why would I not do the comparison test on the bottom one well I might think of guessing that this is kind of like root

Speaker A

N cubed over N to the fourth but these signs work the wrong direction this is not less than that it's not less than the standard BN you might guess it's greater than it I'm saying greater than a convergent thing tells you nothing so I can't do the comparison test even though it's nice and easy I have to do the limit comparison test so if I choose my a.m. to be what I have my BN is highest part on the top highest power on the bottom and you could manipulate this thing I have a 1 over N

Speaker A

to the power of 5 hats I'll note that when you're doing comparison and limit comparison test it's some examples mean you can only use limit comparison test and not comparison some examples you can only use comparison and not limit comparison so be free to use both generally limit comparison test is a bit stronger and applies to a wider set of example so keep that for sure your tool but but you might have to use both of these all right that leaves the ratio test and the

Speaker A

root test so what are we gonna do here now the root test is kind of like the alternating series where there's that minus 1 to the end that you just want to focus in on and say that means I want to use an alternating series well for the root test the big clue here is that in the second one we have ^ n in our expression it's a whole bunch of mess to the power of n and indeed the root test is designed to make those types of series really nice so if you see a whole

Speaker A

bunch of messy stuff to the power of n think root test indeed if I do this if I take this limit of the nth root of it the enth root is like to the power of 1 over n there's always an in there so it cancels and I'm just looking at what the limit of this inside does this goes to the value of 3 halves which is bigger than 1 that you need for the root and so it diverges and then final what I guess is gonna have to be the ratio test just by a process of illumination but

Speaker A

maybe what I would say is that the one I didn't do this last if this was just first the big thing that Clues in for me that is a ratio test is that factorial term factorials weren't really really really nicely with ratios as does a number to the power of n in fact you might think this looks a little bit like a geometric series because of that 3 to the power of n and you might recall that when we talked about what the ratio test was it was trying to relate series to

Speaker A

the geometric series so things that are mismatches like this where there's that 3 the end that might make you think to your metric series but there's also something like an N factorial that screams out ratio test nonetheless you can go and take this limit of the a n plus 1 all over the a n and in this case I'll show you the L true at the end it's equal to zero and so that this limit is gonna converge by the ratio test all right that was me just speeding through

Speaker A

these eight different series not doing all of the work that you would need but just trying to talk about why would I use what method when and I'm just gonna step away and I'm gonna put up the full solution to all of the eight series now [Music]

Speaker A

[Music]

Topics:convergence testseriesgeometric seriesintegral testalternating series testdivergence testcomparison testlimit comparison testroot testratio test

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