Video transcript
- [Narrator] Let's now expose ourselves to another test of convergence, and that's the Alternating Series Test. I'll explain the Alternating Series Test and I'll apply it to anactual series while I do it to make the... Explanation of the Alternating Series Test a little bit more concrete. Let's say that I have someseries, some infinite series. Let's say it goes from N equalsK to infinity of A sub N. Let's say I can write it as or I can rewrite A sub N. So let's say A sub N, I can write. So A sub N is equal tonegative one to the N, times B sub N or A sub N is equal tonegative one to the N plus one times B sub N where B sub N is greaterthan or equal to zero for all the Ns we care about. So for all of these integerNs greater than or equal to K. If all of these things, ifall of these things are true and we know two more things, and we know number one, thelimit as N approaches infinity of B sub N is equal to zero. Number two, B sub N isa decreasing sequence. Decreasing... Decreasing sequence. Then that lets us know thatthe original infinite series, the original infinite series, is going to converge. So this might seem a littlebit abstract right now. Let's actually show, let'suse this with an actual series to make it a little bit more, a little bit more concrete. Let's say that I had the series, let's say I had the seriesfrom N equals one to infinity of negative one to the N over N. We could write it outjust to make this series a little bit more concrete. When N is equal to one, thisis gonna be negative one to the one power. Actually, let's justmake this a little bit, let's make this a littlebit more interesting. Let's make this negativeone to the N plus one. When N is equal to one, this is gonna be negativeone squared over one which is gonna be one. Then when N is two, it'sgonna be negative one to the third power which is gonna be negative one half. So it's minus one half plus one third minus one fourth plus minus and it keeps goingon and on and on forever. Now, can we rewritethis A sub N like this. Well sure. The negative one to the N plus one is actually explicitly called out. We can rewrite our A sub N, so let me do that. So negative, so A sub N whichis equal to negative one to the N plus one over N. This is clearly the samething as negative one to the N plus one times one over N which is, which we can then say this thing right overhere could be our B sub N. This right over here is our B sub N. We can verify that ourB sub N is going to be greater than or equal to zerofor all the Ns we care about. So our B sub N is equal to one over N. Clearly this is gonna begreater than or equal to zero for any, for any positive N. Now what's the limit? As B sub N, What's the limit of B subN as N approaches infinity? The limit of, let mejust write one over N, one over N, as N approachesinfinity is going to be equal to zero. So we satisfied the first constraint. Then this is clearly a decreasing sequence as N increases the denominatorsare going to increase. With a larger denominator, you're going to have a lower value. We can also say oneover N is a decreasing, decreasing sequence for the Ns that we care about. So this satifies, thisis satisfied as well. Based on that, this thing is always, this thing right over here is always greater than or equal to zero. The limit, as one overN or as our B sub N, as N approaches infinity,is going to be zero. It's a decreasing sequence. Therefore we can saythat our originial series actually converges. So N equals 1 to infinity of negative one to the N plus over N. And that's kind of interesting. Because we've already seen that if all of these were positive, if all of these terms were positive, we just have the Harmonic Series, and that one didn't converge. But this one did, putting thesenegatives here do the trick. Actually we can prove thisone over here converges using other techniques. Maybe if we have time, actually in particularthe limit comparison test. I'll just throw that outthere in case you are curious. So this is a pretty powerful tool. It looks a little bit aboutlike that Divergence Test, but remember theDivergence Test is really, is only useful if you wantto show something diverges. If the limit of, ifthe limit of your terms do not approach zero, then you say okay, thatthing is going to diverge. This thing is useful because you can actuallyprove convergence. Once again, if something does not pass the alternating series test, that does not necessarilymean that it diverges. It just means that you couldn't use the Alternating Series Testto prove that it converges.
