Series Convergence & Divergence
This page of videos will cover most of your series needs, so that's why it's so long. Awesome, right?
Intro to Convergence & Divergence (free)
As you probably noticed, there are a LOT of videos in this chapter, most covering some bizarre-sounding "test" for "convergence" or "divergence". Naturally, you're wondering what the heck any of it means, so that's what this video is all about. Getting the overview and concepts before the mind-numbing nitty gritty of the videos to follow!
Nth Term Test For Convergence
This test basically tells you that if your terms aren't approaching zero, there's no way the series converges. Even if the terms are approaching zero the series could still diverge, though, hence the Harmonic Series, so this test isn't super-useful on its own. But hey, gotta start somewhere when delving into the exciting world of infinite series!
Geometric Series Convergence
Don't blow this video off because it's just geometric series and you've seen these before. Because you haven't seen them like this. These ones are infinite. These ones have sigma notation. And this being calculus, naturally there's a special rule just for geometric series that you have to memorize and apply. You'll be glad you did, though, because it turns out these are some of the easiest problems in this chapter if you know how to spot them.
P-Series Test
This video introduces the first of the new series types we'll be working with (and expected to memorize) in this chapter: the P-Series. Though it sounds like it might be an abbreviation for Power Series, it most definitely is not, so be careful. P-Series are more like geometric series, the famous Harmonic Series being the P-series where p=1.
Integral Test For Convergence
Just when you thought there wouldn't be any more calculus in calculus, you get to do some integration! It's pretty wacky integration, though, requiring Improper Integrals, so if you're not rock-solid on those, you might want to review this previous chapter.
Comparison Test
One of the more difficult tests for convergence, this one is annoying because it works for some cases and not others, and it's not plug-and-chug: you have to be a bit clever with picking what to compare your series to. This video will sort most of that out.
Limit Comparison Test
Like the regular comparison test, this one tests for divergence or convergence by comparing two series. On the bright side, this method is a lot more plug-and-chug: once you pick the series to compare, you just throw them into a limit problem and execute. The devil is the details, though, since with this it's a bit trickier to pick the series you're comparing.
Ratio Test
The easiest and most useful test when checking the convergence of nastier problems, in this video I'll show you the key "tells" to watch out for to help you decide when to use the relatively straightforward ratio test rather than the more difficult comparison test.
Nth Root Test
Not the most useful test, but what the heck, most schools and books cover it, so I guess we should mention it here. It basically only applies when there's an exponent of n surrounding the whole problem, which let's face it, is kind of rare.
Alternating Series, and Absolute & Conditional Convergence
Just when they finally give you a type of series that almost always converges -- alternating series -- they have to muck it up by giving you sub-categories of convergence: absolute and conditional. Mean calculus! Oh well, they're still only as bad as the other series, so you just end up using all the tests you've learned up to this point, just with a couple of new vocab words to slap on.