Definition and Notation of Infinite Series by @mes on 3Speak
▶️ Watch on 3Speak - YouTube - Telegram - Summary - Hive Notes - Sections playlist - Sequences and Series playlist - MES Links --- In this video, I go over infinite series (or just series), which i...
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Infinite series and notation. If we try to add the terms of an infinite sequence, remember the notation before was like this, A, N, where N is equal to, you know, one starting point, and then up to infinity, this is an infinite sequence, just keeps going further, like, infinitely, just going A1, A2, A4. So remember, we get an expression of the form A1 plus A2 plus A3. So this time, this time we're trying to add them. Before it was just a sequence, now we're adding all the terms to get a summation. Yeah, adding at an infinite sequence, we get an infinite series, or just series. So A1, A1, plus A2, plus A3, plus A4, etc, all the way plus AN, which is called an infinite series or just a series, and is denoted for short by the symbol, and you could use the sigma right here. I guess, put sigma, infinity, n equals to 1 of A, N, or you could just write this, just in general like this, without that. And this just usually is assumed that you mean this way. All right, so now the question that arises is, but does it make sense talk about the sum of infinitely many terms? Well, first of all, yeah, it would be impossible to find a finite sum for the series, 1 plus 2, plus 3, plus 5, although it plus n, because as you can see, the end just keeps getting larger and larger. Each term gets larger and larger, so it goes to infinite. Because if we start adding the terms, we get the cumulative sums, yeah, we get 1, 3, 6, 10, so 1 plus 2 is 3 over here, and then we have 1 plus 2, plus 3, that's going to be, well, 3, 3, 6, and so on, and you get 15, 21. And after the nth term, we get over here. yes, after the end term, we get n times n plus 1 over 2, that's the sum, which becomes very large as N increases. MES note, recall the proof for this formula for the sum of consecutive integers is shown in my early video here is a link to it. However, if we start to add the terms of this following series, 1 over 2, plus 1 over 4, plus 1 over 8, plus 1 over 16, plus 1 over 32, plus 1 over 64, and you can see the bottom is getting squared every single time. which equals to over here plus 1 over 2 to power of n. So just 2 to the power of 1, this is 2 to the power of 2. This is 2 to power of 1, this is 2 to power of 3, etc. So you're dividing by bigger and bigger number. In other words, it's going to 0. So the, yeah, but the summation is going to be, in this case, finite. The following table shows that, shows that as we add more and more terms, these partial sums, remember these are all just partial sums, these ones here's 1, 3, 6, 10, et cetera, partial sums, become closer and closer to 1. Yeah, and it's another MES note right here, note one. This is similar to my earlier video on Zeno's paradox in which walking to any given wall is, quote, impossible, since you need to first walk half the distance at each interval, thus never actually reaching it. Hence the idea of, of quote, limits arises, and you just a link to that. So yeah, if you're trying to walk to a distance, let's say 10 meters, you need to first walk 5, and then you need to walk 2.5, and then you need to walk 2.5 by 2. So every single time you need to go to half. arises a very interesting paradox. So now, and also MES note to the following Excel spreadsheet file link will be used throughout this video, so make sure to download it and play around with it. So I'll be using this one whenever I reference it. So if we take these terms right here and go to just add up the terms. So you can add them up manually using Excel or just using this formula right here to get this. And notice what happens here. So here's the terms. So the term 1 over at 2 to the end, so initially you have 0.5, which is just 1 over 2 to the power of 1, which is 1 over 2. It's 0.5, sum of the terms is 0.25, then the second term will be 0.25, the sum of these two. And then as you get further and further, notice what we have, this is going to close, this is getting closer and closer to 0. This is going all the way to zero. Look, the decimal place is getting more and more zeros. And then as this one's getting more and more zeros, this one's getting closer and closer to one. This is going closer to zero. So that's very interesting. So in fact, by adding sufficiently many terms of the series, we can make the partial sums as close as we like to want. So you could add as money as you want, and it will always get closer and closer and closer. This just gets more and more nines. Yes, so it seems reasonable to say that the sum of this infinite series is 1 and to write, so yes, it seems reasonable that we could just write this infinity right here and equals to 1. So the summation of 1 over 2, the power of n, which equals to 1 over 2, the power of 1, 1 over 2, 1 over 4, plus 1 over 8, plus 1 over 16. equals to, or it keeps going on, let's go plus dot, dot, dot, plus 1 over 2, for completeness, this equals to, well, put another dot, dot, dot, so you can keep going on and on and on, this equals to one. So it seems reasonable to state this. And going further, we could use a similar idea to determine whether or not a general series has a sum. So consider the following partial sums right here. S1 equals A1, S2 equals A1 plus A2, and S3 is equal to A1 plus A2 plus A3, and this one's S4. So, let's fix that up. So S4 equals to A1 plus A2 plus A3 plus A4, and in general, what we have is the nth partial sum equals to A1. Yeah, a1 plus a2, plus a3, plus dot, dot, plus a n. And we could write this as equals 2, as a partial sum n, I equals to 1, and then A1. So this is a partial sum as opposed to this infinite sum like that, symbol there. So this infinite sum, and then we have a partial sum. And now these partial sums, they form a sequence and we'll write this as SN, or this is lowercase like that. So yes, the sequence is just a number of terms, or just the terms, and then series is addition. Remember that. And, yeah, these partial sums, which may or may not have a limit. So if the limit as n approaches of infinity, as n approaches infinity, as n of the partial sum equals to s, exists, so if this exists as a limit as an actual finite number, then, as in the preceding example, where approaches 1, we call it the sum of the infinite series, and this is the summation right here, A, N. So the sequence is like this, and the sum is the series. And now we'll get to the definition 1. So given the series right here, the summation as from n equals 1 to infinity of A.N, so the infinite series of this infinite sequence of equals to A1 plus A2 plus A3, let S.N. denote its nth partial sum, and we'll write this as, so the nth partial sum, as I wrote above, and let's write it here in the definition. So from i equals to 1 up to n of a i equals to a 1 plus a 2 plus a, yeah, plus dot dot, plus a n. So that's what it is. Yeah, so if the sequence SN of the partial sums is convergent and limit as n approaches infinity of s n equals to s exists as a real number, then the series right here, the summation of the sequence AN is called convergent, and we write, so if it converges to a number, for example, 1 as in the above example, so A1 plus A2 plus dot, dot, plus a n, and it just keeps going on all the way to infinity, then this equals to us. So if it equals to s, or we have like this, infinity, n equals to 1 of a n equals to s. So it's convergent if the limit goes to a number s, like that. And note here that this A-N, this is AI, this one's because it's a partial sum, this one is we're taking the limit as it goes to infinity, whereas this one is going to be just A-I all the way to whatever N is. So the number S is called the sum of the series, and otherwise the series is called divergent. So if you can't get a number, a real number that exists as the limit as this approaches infinity, as the summation approaches, or the partial sums approach infinity, or the N approaches infinity, the number of terms in the sequence. So if you can't get a real number, then it's called divergent. So going further, so thus the sum of a series is the limit of the sequence of partial sums. Yeah, so you take the limit as these partial sums goes to an infinite number of partial sums. and if it's approaching a number, a limit, then the series is convergent. So we write the sigma right here, summation of from n equals to 1 to infinity of A, n equals to S. So when we write that, we mean that by adding sufficiently many terms of the series, we can get as close as we like to the number S. So it doesn't matter how many terms we can add up, they will always get closer and closer to us. And also notice that, and just to write what I've been saying for the past few minutes. So the summation or sigma, A of infinity, n equals to 1 of A, n equals to, yeah, this infinite sum like this, is the same thing as writing, well, limit as N approaches infinity of the partial sum, summation, of I equals to 1, A, I, N, like that. Yeah, so the infinite sum, to write it in a limit, you would write it like this. So there's a partial sum, as this n goes infinity, we get the full sum. And now to compare this with the improper integral, just to illustrate it further. So if you have something like an integral from 1 to infinity of F of X, like this, this equals to, well, you can't just do a deal with infinity like that, so you need to take a limit. So whenever you're in infinity, you need to take a limit. So the limit is going to be limit as n approaches infinity, or actually a T. Let's use T for this one. So as T approaches infinity of, from one, integral from 1 to T of F of T, DT, or actually I had no need to change a variable, just leave the X like that, affects the X. Yes, just to remove confusion, choose the same one. So, essentially, if there's infinity, you just need to take a limit, just change a variable right here, or just to pick a variable T, like here. This one's I, and then as I... approaches n where n is approaching infinity so yes fascinating fascinating and yes to find this integral we integrate from 1 to t and then let t approach infinity as this limit right there so for a series we sum from 1 to n then let n approach infinity so here we sum from 1 to n and let it go to infinity