[Script Info]
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[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,One type of series where we can \Nactually come up with a formula
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,for the sequence of partial sums
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,is called a geometric series.
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,You might remember learning \Nabout geometric sequences
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,in the previous section.
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,Basically, it's the same idea except \Nthe terms are being added together.
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,Every term is being multiplied \Nby a common ratio, which we call r,
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,and that means the formula \Nfor our generic nth term would be ‘a’,
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,which was the first term,
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,and then times r to the \Nn minus 1st power,
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,so that would be the formula for \Nour series for each individual term.
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,But the idea here is, in some cases,
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,we can find a formula \Nfor the nth partial sum,
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,for the sum of the first n terms, \Nand based on that,
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,we can determine whether \Nthis series will converge or diverge.
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,A couple of cases to get \Nout of the way first,
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,if r is equal to 1, \Nwhat would this thing look like?
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,Well, what it would look like is \Njust ‘a’ and then plus another ‘a’
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,and then plus another ‘a’ and \Ncontinuing in that fashion forever.
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,Now, if I just added up \Nthe first n terms here,
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,I would have a total of n a’s \Nall being added together.
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,That would be n times ‘a’ [na].
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,But if we take a limit as n \Napproaches infinity for n times ‘a’,
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,that limit is going to be infinite.
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,So if r is equal to 1, \Nthis series is going to diverge.
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,Hopefully, that makes sense;
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,if you add up an infinite number of \Nterms and they're all staying the same,
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,your sum is not going \Nto approach a value.
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,Basically, the only way \Nthis wouldn't diverge
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,is if ‘a’ is equal to 0,
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,and most of the time, \Nwe're going to ignore that case
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,because it's really not a very interesting \Nversion of a geometric series.
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,Next up, we're going to look \Nat the case: if r is not equal to 1.
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,In this case, the sum of the first n terms
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,would just be all of these \Nterms added together.
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,So this is our nth term.
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,But actually, we can do something \Ninteresting algebraically here
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,if directly below this, \NI take r times all of the terms,
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,because if I do that, \NI take r times this term,
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,I'm going to get ‘a’ times r.
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,If I take r times this term, \NI'll get ‘a’ times r squared.
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,I'm going to continue in this fashion,
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,and this is actually going to \Nbe the second-to-last term
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,because that's what I would get \Nif I took the previous term times r.
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,That means, if I take this times r, \NI'm going to get ‘a’ times r to the n.
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,And it turns out, since both \Nof these sums are finite,
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,(there's a set number of terms),\NI can combine them together.
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,But surprisingly, \NI'm actually going to subtract them.
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,I'm going to get Sn minus r times Sn.
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,My goal here is to come up \Nwith a formula for just Sn.
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,But the interesting thing that happens is,
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,if we were to subtract all of \Nthese terms from these terms,
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,almost all of the terms \Nare going to cancel out.
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,The only ones that are going \Nto be left are the very first term
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,and the very last term of this one.
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,Since we're solving for Sn,
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,it hopefully makes sense to factor \Nthat out from this side of the equation.
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,There's also a common factor of ‘a’
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,that we could factor out \Nfrom this side of the equation.
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,Solving for Sn, we can get this \Nby itself by dividing by 1 minus r.
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,The reason we don't have \Nto worry about dividing by 0
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,is because we're working \Nunder the assumption
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,that r is not equal to 1 in this case, \Nwhich gives us this formula.
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,What we're looking at here
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,is a formula for the sum of the first \Nn terms of a geometric series
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,as long as our common ratio is not 1.
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,What I'm interested in, though,
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,is if the infinite series is \Ngoing to converge or not.
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,Since we have a formula for Sn,
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,now we're going to take the limit \Nof this as n approaches infinity
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,to see what's going to happen to the sum.
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,One thing you'll maybe notice here is,
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,there's only one part of this\Nentire formula that has an n in it,
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,and it's this part right here.
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,This is actually something \Nwe talked about in Section 11.1.
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,This right here is only going to converge \Nif the absolute value of r is less than 1.
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,Think about it: If this is greater than 1,
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,then multiplying it \Nby itself multiple times
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,will cause it to basically \Nblow up to infinity.
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,And if it's less than negative 1, \Nthe same problem is going to happen,
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,except the sign will be alternating \Nfrom positive to negative.
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,This is only going to converge \Nif the absolute value of r is less than 1.
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,Any other value for r, \Nand this thing is going to diverge,
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,except of course, 1, \Nwhich we've already ruled out.
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,Basically, this sequence, which stands\Nfor the sequence of partial sums,
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,converges if the absolute \Nvalue of r is less than 1,
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,and it's going to diverge if the absolute \Nvalue of r is greater than or equal to 1.
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,We've already excluded this case,
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,but if r was negative 1, \Nwe would run into the same problem.
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,It would still diverge.
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,What does it converge to?
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,Well, if the absolute \Nvalue of r is less than 1,
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,then this term right here is \Ngoing to approach 0 as n grows.
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,And that means, \Nwhat we're going to be left with
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,is just ‘a’ in the numerator \Ndivided by 1 minus r,
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,and that winds up being \Nthe sum of a geometric series.
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,If the absolute value of r is less than 1,
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,then this series (the geometric series) \Nis going to have a sum,
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,and that sum is going to equal \N‘a’ divided by 1 minus r.
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,If this condition is not met,
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,then the geometric series \Nis going to diverge.\N