 In this video, I want to talk about some properties of infinite series. So suppose we have two series, the sum of the sequence A and the sum of the sequence B, and suppose these two series are convergent, so we're assuming that the series is convergent. Then it turns out that if you add together the sequence, the associated series will be convergent. Or if you were to take the series, sorry, if you take the sequence A and you times it by some scale of C, then the associated series will likewise be convergent for any constant. Or in slightly more expanded form, you see these things right here. The series of C times A to the N, this is the same thing as C times the series of A, N. And hence, convergence will be the same for these two things. Or if you take the series where you add together a sequence, or even if you subtract the sequence, this series will be the same thing as the series of the A's and the B's separately and you'll add together those series. Now this right here is commonly referred to as the linearity property of series. And this is a property that we've seen many times in previous calculus settings. Limits have this linearity property. That is to say, if you add together limits, are you scale limits? That's all you have to do to calculate the limit. The sum of limits is a limit of sums, things like that. It turns out that derivatives were linear, anti-derivatives were linear, integrals, both indefinite integrals, finite sums are linear. Now we see that infinite sums, aka series, are likewise linear operations. So these are linear operators. Now I make emphasis on these linear operators here because calculus uses a lot of linearity. This idea of linear algebra shows up all over in calculus. And so if anyone is interested in learning some more about the importance of this linear algebra, I would encourage you to actually learn some more. You know, take a class of about linear algebra or you could take a look at my channel about an open source linear algebra textbook to find out some more about these things. Now I should make mention that it's very important that in order for these properties to hold, we are assuming that the two series are convergent on their own. Because it turns out that if you're not careful, you can combine two divergent series by addition and subtraction actually form a convergent series. So for these linear properties hold, we are assuming that all the series are convergent on their own. So an example of how we can use these properties to help us determine convergence and actually compute the sum of a series, consider the following. Take the series n equals one to infinity of three over n times n plus one and one over two to the n. Now we could break this series up into pieces, right? We could take the sum of three over n times n plus one, where n goes from one to infinity, that we take the first series and we separate it from the second one, n equals one to infinity via one over two to the n. And we treat these two separately. And then also, we notice that that coefficient of three on top, we could actually factor it out. We get a three out in front and the one right here. And so again, we can use this linearity to kind of break these things up. Now this first series right here is an example of a telescoping series that we've actually considered this exact one in a previous video. This telescoping series, you take the one over n times n plus one, you can break this up into one over n minus one over n plus one. And then when you take that, what's going to happen is that this series is going to turn out to be one. We computed this previously. Now the second series, this one right here, is actually an example of a geometric series. Likewise, we consider this one in a previous video. It's a geometric series. This is a geometric series where your first term a is equal to one-half, use plug in n equals one, and your constant ratio is also one-half. And so using your formula, this should be a over one minus r, this right here will turn out to likewise be one. And therefore, when you use the linearity properties with these series computed individually, you'll get three times one plus one, this series will add up to be four. This gives you the sum of the series here. So we can use properties of series to compute more complicated series by comparing it to simpler and smaller series. One last important property I should mention about series in this video is that a finite number of terms does not affect the convergence of a series. What I mean by that is consider the following example. Take the series where n equals one to infinity, like so, and then our sequence will take n over n cubed plus one here just as an example. Now if we were to start expanding this thing, the first term in this in the sequence n over n cubed plus one, that's equal to one-half. The second term is two-ninths. The third term is three over 28. And then if I just kind of stop there, we often say like dot dot dot. What that means is we actually have a series where n equals four to infinity and the exact same sequence n over n cubed plus one. So we can see that as you take the sum from one to infinity, this will just be the first three terms plus the series from four to infinity. And so it follows here that if the series that starts at four is convergent, that is this series adds up to be a finite number, then we're going to take a finite number and add to it three more finite numbers and therefore the sum of all these numbers is likewise finite. So that would imply that the series starting at one is likewise convergent. So if the one starting at four is convergent, the one starting at one has to likewise be convergent. But it turns out we can actually flip this direction around if we start the series at n equals four and go off towards infinity, same sequence in over n cubed plus one. There's really no significance about the sequence here, this is just an example. This is actually equal to the series which starts at one, n equals one to infinity, n over n cubed plus one. But we actually have to subtract a few things, we have to subtract one half, we have to subtract two nights, and we have to subtract three over 28 to compensate for that. And so we see the exact same thing happening here. If we know that the series starting at one is convergent, if we knew this one was convergent, that means this series is a finite number. In which case if we take a finite number and we subtract three finite numbers from that, that's going to be a finite number. Thus this implies that the series starting at n equals four is likewise convergent. And in this example here, there's nothing significant about the sequence in over n cubed plus one, you can do any sequence and this would still be true. And starting at four here also is some of an arbitrary choice. If it's convergent at four, it'll be convergent at one. If it's convergent at one, it'll be convergent at four. And any other number as well, the starting value doesn't actually have any significance in terms of the convergence or divergence. And I should also mention that if you'd have, if this series here was divergent, right, that if this series was divergent, the one starting at four, well the ones, there's one starting at one, if this thing were convergent, then you subtract the number seven forces convergence over here. So it turns out that if these, if one of them was divergent, the other one has to be divergent. If one of it's convergent, the other one has to be divergent as well. So the convergence will be the same irrelevant of the starting number. So we'll see lots of formulas that start at one or start at zero, but be aware those can be modified to four or 17 or 5,280. Doesn't make any difference. The convergence is independent of the starting location. It depends on the sequence not really the starting number there.