>> Now when, when items in a sequence are added, the situation is called a series. And a series can be either infinite or finite. That is, we can be adding an infinite number of numbers, or we can be adding a finite number of numbers. Here, we're adding a finite number of numbers, and so it's called an emph [phonetic] partial sum. An emph partial sum. OK. Let's take a look at a couple of other situations. Here, we have, let's see, now, the counting mechanism here, the, the index is N rather than K, and it can be pretty much any letter that you'd like to assign to it. The lower limit is 1, once again, and this lower limit can change. It can be almost anything that you want it to be. It could be 5, and then we can go to 10 or, or whatever. So it can be all over the board, but at any rate, the lower limit is 1, the upper limit is 3 just as was the case in the previous problem. The expression is 4N. So N is taking on values from 1 to 3. So we have 4 times 1 plus 4 times 2 plus 4 times 3. In evaluating, we find that sum to be 24. Now one of the properties involved in, in this kind of notation is this. You may notice that 4 is a factor in all three of these terms, and it does turn out that because of this, the nature of this expression, that 4 can be written over here. That is, we can take the 4, and write it as a coefficient of this summation notation. You see, we can just kind of alter it like this, and we get the same result because this means we're going to add as N takes on values from 1 to 3. So it's 4 times, now this part is 1 plus 2 plus 3. So it's 4 times 1 plus 2 plus 3, or 24 as expected. Now another, another property has to do with the notion that summation notation, or this summation idea, can be distributive, or it is distributive in nature. Let me show you an example of that. Come back over here, and, and let's look at this, this first problem. I could actually rewrite this like this. I could say, well, the sum, the summation of items given by this expression can be thought of as the sum of this rascal plus the sum of that. That is, it's the sum as K runs from 1 to 3 of 2K plus the sum as K runs from 1 to 3 of 1, and this could further be manipulated a bit. It, it really isn't necessary in this context because the problem is pretty simple, but in other situations, this is going to be a pretty valuable tool. The 2 can become the coefficient here. So we could write this as 2 times the summation as K runs from 1 to 3 of K and then plus the sum as K runs from 1 to 3 of 1 and evaluate these separately. Now, this, you know, we, we just saw an example of this situation where, where K takes on the value 1, 2, and 3. So we'll have, this'll be 2 times and then 1 plus 2 plus 3. And we haven't seen this situation yet. We're going to talk about a little more over there, but to brief you on this, it, it's this idea. That what is the value of this expression when K is 1. You see, the expression has no K in it. It, it won't change value at all as, as K changes value, the expression here won't change value because there's no K in it. So when K is 1, this expression is 1. When K is 2, the expression is 1. When K is 3, the expression is 1. Well, at any rate, adding here, we get the same result that we did before. Alright. Here's another situation. Now, in this one, the counting mechanism is starting at 0, and it goes to 3, and the counting mechanism is part of an exponent. It is the exponent on 2. So we would have, let's see, now, so 2 to K as K goes from 0 to 3. So it'd be 2 to the 0, plus 2 to the 1, plus 2 squared, plus 2 cubed, and then evaluating, we find the value to be 15.