In this video, I go over Theorem 2, which states that we can move a constant out of the sum of a series as well as add or subtract series individually. These follow from the Limit Laws of Sequences...
Transcript
Theorem 2 states if the series A-N and B-N are convergent series, so they sum to an actual value, then so are the series, this right here, C-A-N, so where C is a constant, and the addition of the series, A-N plus B -N, and the subtraction of the series is also, is also convergent, and we're given here, and we have these terms that are true as well, I, so we have, summation of n equals to 1, up to infinity of C, A, N equals to, we could just take the C up, just a constant like that. Infinity, n equals to 1, A, N. And likewise here, I, I, I, we have, if we have the addition of the two series, A, N equals to 1, and then we have like this. a n plus bn we could just break these up and these will be like this infinity n equals to 1 a n plus infinity n equals to 1 bn and then obviously we'll have a subtraction as well where we can separate these series out and like and equals to infinity of a n minus bn and this equals two, yeah, summation, and, uh, yeah, of n equals to 1 to infinity of a n minus. like that, infinity n equals to 1 of bn, like that. So, yes, these properties are this theorem 2, these properties of convergent series follow from the corresponding limit laws for sequences from my earlier video on infinite sequences. Link was earlier above. For instance, here is how part 2 of theorem 2 is proved. So we want to prove this addition one here, how you could separate the series. What we can do is, well, let, let's let S.N. equals to the summation or the partial sum n i equals to one of a i and then s to oh then s is a total sum of the series so n i or this is just going to be n uh n equals to one of a n like that so there's a total sum we have infinity this was the partial sum s n and then for the second uh series b remember they're both convergent What we get here, I'm going to call this TN for the partial sum. TN equals to summation N, I equals to 1 of BI, like that. And then we have T for the total sum of the B series. This is approach to infinity like that. of bn. So what I'll do now is a line like this. This is total sum as opposed to infinity. And here is the partial sums. It's just partial sums, total sums like that. All right, so now that we have those terms defined, we can do is the nth partial sum for the series A-N plus B-N. So we want to write this combined. It's going to be, and I'll just use U instead. I'll say U for together you've unified so u n partial sum is equal to this is up to n i equals to one of the partial sums like that or it's going to be partial sum so of a i plus b i like that and now taking the limit and breaking the sum into two parts is using summation laws we have a limit as n approaches to infinity Because again, we're dealing with this infinity, you have to be more formal, and we're always right limit. I guess a limit as an approach to infinity of U.N equals to limit as n approaches infinity of the summation. This is n i equals to 1 of a i plus b i, like that. And now that we have a limit like that, what we could do is, well, write this down as limit. as n approaches infinity and then break the sum into together so this is going to be a i this is n i equals to 1 plus like this n i equals to 1 of b i like that and then what we can do is break the limit inside both of them like that this equals to limit as n approaches infinity like this of summation a i and i equals to one plus and yes so it's pretty straightforward but just want to be more formal as my calculus book has gone is for dealing with infinity so n approaches infinity of n i equals to 1 b i like that so therefore what we have here this is limit as n approaches infinity of this one here a i from i is equal to 1 to n but notice here you notice here this is just our s n and our tn so what we're doing here is taking the limit this equals to limit as n approaches infinity of s n plus limit as an approach is infinity of the partial sum TN. And we're taking the limit of these partial sums to get the total sum. So we get S plus T like that, where S plus T is just these over there. So thus, so therefore the addition A& plus B in the series is convergent because, well, these are convergent. S and T are convergent, like this, this is conversion as we're given above here, if so, if AN and BN are convergent. Yeah, so that's what we end up having is the summation, uh, like this, infinity n equals to 1 of a n plus bn is equal to because this is just Yeah, over here, UN is equal to this, partial sum of the addition, but now we're taking the limit to get the actual one, so summation from N is equal to 1 to infinity, A-N plus B-N, this just equals to S plus T, which equals to, again, S plus C, those are just a total sums, put infinities over there, this equals to infinity, n equals to 1, infinity n equals to 1 bn like that and this is just exactly the theorem states here but again because if we're dealing with infinity I just wanted to go over in a more formal way even though this yeah this is pretty straightforward but all I'm doing is formally writing the limit n equals infinity of using this partial sum formation i equals 1 up to n then show that you could break up these and then put the limits in, and then you'll just get exactly what you were looking for. So it's fascinating stuff. Alright, so now let's look at an example that utilizes this theorem there. So example nine, find the sum of the series, and now we're going to be using a lot of what we've learned earlier or just utilized earlier. So find a sum of the series from n equals 1 to infinity of the summation. right here, 3 over n times n plus 1, and then plus 1 over 2 to the power of n. So first off, note that the series, if you look at this separately on this side, right here, 1 over 2n, 1 over 2 to the power of n, is a geometric series with A equals 1 1⁄2 and r equals 1 1⁄2. So, let's write this out for completeness. We have n, R, go infinity, like this. infinity of n equals to 1. 1 over 2 n equals 2. I'll plug in 1, so 1 over 2, then plus 1 over 2 to the power of 2, plus 1 over 2 the power of 3. Yeah, you keep getting more and more n, so 2, 2, and 3 and 4, et cetera, plus dot, dot, dot, so we have a is equal to 1 half. and then here it's separated by the so r is just one half as well so you can see here one over two half times it by one over two like that equals to one over two one over two for completeness two and this adds up two three like that so this is our r and remember the geometric series the sum is just a over one minus r so which equals to one half over 1 minus 1 half. Yeah, so 1 minus 1 half is just 1 half. So we're left with 1 half over 1 half, this all cancels equals to 1. 1.5 divided by 1⁄2 equals to 1. So we know the sum of the second part, and we could use the theorem, and we'll be utilizing that soon. And now for the first part of this summation there, or of the series, we already saw this one in an example 6 using that telescoping cell. So in example 6, we found that the summation, I'll write this, infinity n equals to 1 of 1 over n plus 1 like that, this equals to 1. So we know that, and what we can do is just take this constant out, separate them, add them up, and we get the answer. Yeah, so by theorem 2, the given series is convergent because the parts in between are convergent. So 1 and 1, so circle out. So that's going to be 3 times 1, it's going to be 3, and then we could just add these up. So if these are convergent, scrolling up here, if A and B are convergent, then so are the series and the summations are at the constant or the difference. There, so let's go over here, so thus what we end up having is the summation, infinity, so infinity like that, n equals to 1 of 3 n plus 1 plus 1 over 2 the power of n equals to, we can just separate this out and move the constant out, so take the 3 out, and then we have like this, because yes, we're dealing with infinity now, n equals to 1 of 1 over n, and plus 1 like that, plus summation n equals to 1 of this geometric series 2 to the n, and this just equals to 2, well, yeah, this just equals to 3 times 1 plus 1, equals to 4. So, yes, fascinating, fascinating stuff like that. So it's convergent equals to 4. And now make another note, a finite number of terms doesn't affect the convergence or divergence of a series. For instance, suppose that we were able to show that the following series is convergent. So if we had, if this is convergent, n equals to just doesn't matter how many, find a number of terms. So if you have n equals to 4, so we start off there, of n over, n cubed plus 1. So yes, if we were to show that this is convergent, then it doesn't matter where we start off from, because since what we end up having here is if we had n equals to 1, we just end up having here, put a summation, infinity, n equals to 1, this is going to be n over n cubed plus 1. So when n equals to 1, we have 1 over 1 plus 1 is 2. and then when n is 2, we have plus 2 over 2 to power 6 is 8, plus 1 is 9, and then plus 3, the third one is 3, 3 to the power of 3 is 27, plus 1 is 28, and then we have, so that's the first 3 terms, then we have n equals to 4 there, plus the summation now, infinity of n equals to 4, yeah, like this, n over n cubed plus 1. So yeah, so yeah, it doesn't matter how many terms we have, because if we're given that... This is convergent. So this is convergent. All we're going to be doing is adding a few more terms to get this one here. So it follows that the entire series from n equals 1 to infinity of this is convergent. Similarly, if it is known that, or just in more general, so if it is known that the series from n equals capital n plus 1 to infinity, A.N converges, then the full series is also convergent. And we'll write out like this. I'll write it out better. infinity n equals to 1 of a n equals to the series a n from n equals to 1 up to capital n plus here infinity and this is n equals to capital n plus 1. So if this is convergent, this whole thing is convergent then. So if this is convergent, if then this whole thing's converging because all you're doing is adding a few more terms over here from n to uh yes from n equals 1 to n because we're not dealing with infinity yet because if this is the convergent that's it you're done everything is convergent going further and yeah here I wrote that down so uh if if this is convergent this part here then this is also this whole thing is also converted