Calculus: The Alternating Series Test

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Sequences and Series
The Alternating Series
The Alternating Series Test Page [1 of 2]
Hey, let's take a look at some infinite series, see if they converge or diverge, why not? All right, how about this? . Okay, now the first thing I noticed is, that whenever you see a (-1)^n or (-1)^n+1, that means that all that does is put in a plus, minus, plus, minus, plus, minus. Or, minus, plus, minus, plus, minus, plus, so it alternates. This is immediately visible to me, because this is an alternating series.
So, I'm going to try to jump right into the alternating series test which says, "If you have an alternating series, if you look at the terms without the alternation, if those terms are all positive, decreasing, and the limit of the terms themselves is zero, then you can automatically say that the infinite series converges." So, basically, if you think about it, what we're really saying here, is if you forget about the negative signs, if the terms are shrinking down to zero, the limit converges. This is actually the only time you can really do this. Remember, that if the terms shrink to 0 in general, you cannot conclude the series converges, because they might not be shrinking to 0 fast enough.
The power of the plus or minus alteration allows us to conclude that I jump and then come back a little bit, but not as much as I went, then jump again, jump again. And, it's like a ping-pong ball literally hopping back and forth, but actually heading toward a target. So, the alternating series is a very special type of series. It's a series where as long as the terms go to 0 and they're shrinking, the thing does converge. So, it's sort of like, you know, our fantasy comes true. The terms should go to 0, that' sort of the end of the story. That only works for alternating series. I don't know, adopt this strategy for other series. For example, summation of . The terms are positive decreasing and going to 0, but that series diverges to infinite, the harmonic series. But, if you alternate it, then you're fine.
Anyway, let's take a look at the alternating series that is comprised of . What happens if you just look at the non-alternating part? Plainly that's positive, it's also decreasing as I make n bigger, this thing actually shrinks. And the limit as n goes to infinite of that thing, is easily seen to be 0. So, in fact, all conditions are satisfied. This thing converges by alternating series test.
Notice again, how I like to give a little rationale for why something converges or diverges. So, the whole world knows I'm not just guessing but, in fact, it's an informed answer. . Again, I immediately see this is alternating. I consider the non-alternating part positive, that's clear. It's decreasing as the n gets bigger this thing shrinks. And what's the limit? Well, the limit, in fact, is 0 as n goes to infinity. So, in fact, this automatically converges. Alternating series are really easy, right? Just immediately say, "convergence by alternating series tests." Isn't that great? So all the big series are sort of our friends, because all you got to do is make sure terms are decreasing, going to 0, bing and you're done, that's all there is to it.
Let me just point out, that the alternating the plus or minus thing is so critical, because look at what would happen if we would just take away the plus or minus jiggling. If we take away the jiggle, what do we get? If we take away the jiggle, so no jiggle, we have that. But that's a p-series and the p is 1/2, which is less than 1. So, without the jiggling, the plus, minus, this thing diverges, this will diverge. However, when you allow me to jiggle - so I'm allowed to go backwards, forwards, backwards, forwards and I sort of come in, then all of a sudden, the jiggling makes it converge.
But, I really want you to be careful, even though some series diverge when you make them alternating, they actually could converge. The alternating series are really sort of a peculiar little selection of people. Let's try one last one together. .
All right, let's take a look at this thing. I see that it's alternating, so let me just take a look at this piece right here. If I want to explore that thing and, well, what do I see here? What's the limit? First of all, is this thing going to be decreasing? That's a good question, is it decreasing? Well, the answer is, "Yes, it is decreasing." And, what happens if you take the limit? If you take the limit, it then goes to infinite. Well, then what do I get? Use H'opital's Rule and you see 1/3, and 1/3 is not 0.
So, in fact, this limit must diverge. In fact, this might have been a good place to actually use our "quicky test." Because if you just take the limit of the terms themselves, that's not shrinking to 0. The limit of the terms themselves, in fact, doesn't exist, because this is not going to approach 1/3, but when you put in the plus, minus wiggling, this is going to wiggle between +1/3, -1/3, +1/3, -1/3. So, you're going to be heading toward two different targets at once, +1/3 half the time, and the other half of the time -1/3. So, that limit is not 0.
So, in fact, here is an example where the "quicky test" comes in to rescue us right away, and we see since the terms aren't even shrinking to 0, there's no hope of this converging; this diverges. So, don't forget, even with alternating series, like with any series, always use the "quicky test." Take a look and see if the terms themselves are going to 0. Here's an application, where even though it's alternating, you can stop right there, ding, divergence. All right, great, alternating series, no, no, no, huge deal. Make sure the terms that aren't alternating are all positive, decreasing, the limit is 0, you're home free. Biggest mistake, to take that method and apply it to non-alternating series. Don't do it, don't go there. Think of the harmonics series, , sum it up, it's infinite.
See you at the next lecture.