>> Let's look at both of these. Let's start with this one. Now we can begin the investigation just by thinking about one of the limits of integration as being a variable. Let's just start that way. Let's just say that we want to integrate from one to the variable being of this expression. Now we, we integrate here, the integral function is this. And from one to B and then F of B is this. You see just replace the X with B. Minus this expression and replace X with 1. All right so that's the idea. Negative, negative makes this positive one over one is 1. So we have 1 minus 1 over B. So we can investigate the integral by choosing various values for the B. In other words, we can do this. We could set up a table of values and we can say well when B is 2, we would have 1 over 2, oh that's one half. 1 minus one-half, oh this would be a half. So the integral is one half. If B is 5, we have one minus one fifth. We find the integral to be point 8 and so on. We could investigate for a number of values of B. Then we could, we could see how that integral may be approaching a particular value. And we can even generalize it. We can say gee what happens when B becomes infinitely large? If B becomes infinitely large, then this fraction approaches zero. You see? So as B approaches infinity, this fraction approaches zero and so the integral approaches one. Oh, that's the limit idea. So we could turn this whole notion into one where we just say, let's just turn it into a limit expression for evaluation. And here's how that goes. If we're talking about integrating from 1 to infinity, then we would say well gee just this part right here says that here's what we can do. We can say that we're going; this becomes the limit as B approaches infinity of the integral from 1 to B. You see, so this is replaced with this idea. All right now once we have this, then we go, we slide our thinking over here, ignore this for the time being and deal with this in the way that we have in the past. Now, this becomes the one minus 1 over B that we saw over here earlier. And now we evaluate the limit of this expression and as B approaches infinity. As B approaches infinity, this approaches zero and so the limit is 1. And the graph of the situation looks like this. The graph looks like this kind of thing and we're going from 1 to infinity. And the value is a finite number. And so we say that this integral converges. There is a finite area involved under this notion of converges. Now the other situation, integrals like this either converge or they diverge. If, if a limit, excuse me. If an integral diverges, it means that there is an infinite area. Now, it, it's kind of remarkable I think that some of these situations that look very much alike have very different outcomes. Sometimes we have a situation that looks like it's going to converge and it diverges. Looks like it's going to diverge and it converges. You see, we really don't know. Let me show you a good example. And this one is in your textbook. But here's the graph that we were just talking about, the graph of the situation we were just talking about. Now, let's, let's turn our attention up here. Here's another problem, a problem very similar to the one before. It's 1 over X here. And over here it was 1 over X squared. The graph looks remarkably the same as the other one. But look at the outcome. Alright, first of all we just march through as we did before. That is we change this situation into the limit of an integral, you see? So we're concentrating here, oh we are going evaluate this as the limit as B approaches infinity of the integral from 1 to B. You see, this is replaced with this. And now slide your attention over here. And now how do we investigate this? Well, the integral function of 1 over X is LNX and the integral 1 to B. Ok, so it's LNB minus LN1. But LN1 is zero. What is LNB? What's the limit as B approaches infinity of LNB? Well, as B becomes infinitely large, the natural law of rhythm of that infinitely large number continues to increase. And therefore, infinity then is our answer here.