 Hello, this is Professor Steven Nesheva, and I want to tell you a little bit about Taylor expansions, and in particular the Taylor expansion of this quantity 1 over 1 minus x. So what I've done is I've laid out a set of x values, and now I'm going to calculate that function and make a graph of it. And so here we are. So this is the function 1 over 1 minus x as a function of x, and it's this increasing function. So that's what we want to try to approximate with the Taylor expansion. So the complete Taylor expansion of that function turns out to be really simple. It's just 1 plus x plus x squared and so on, but we don't want to just carry out that expansion forever. We we want to see where we can truncate it. And so the first place to truncate it would be right after that 1, and that's the 0th order approximation. So we would say f of x is just about equal to 1, and let's see how that looks. Well, not so good. You know, that's just the 0th order Taylor approximation, where f is 1. It doesn't do very well. Let's try the second term. That is to say first order approximation has us include x there. And what does that look like? Well, that's obviously the equation for a straight line green here, and it does quite a lot better here for small x, but then starts to to diverge from the real function, which is which is again that black line. What if we add it on the second order term? Well, that's now that's that's blue here. That's 1 plus x plus x squared. It's got some curvature. So it does a little bit better. So we could just carry on with that. And as long as x is smaller than 1, then we are actually guaranteed eventual conversions. But where should we stop from a practical standpoint? For a physical scientist, the answer is often something like when experimental uncertainty makes it pointless to go on. So let's suppose we had some uncertainty in the value of our predicting this function, 1 over 1 minus x, and there are the error bars. So if that's how good we need to be, then obviously red, the zeroth order doesn't do very well at all. Green, the first order approximation does well until about 0.2. Blue does pretty well up until, let's say, with an experimental uncertainty up to about 0.3. So depending on our uncertainty, that determines whether we should take it out to the that approximation or that approximation. So finally, what is the function that you're trying to approximate? Isn't quite in the form 1 over 1 minus x, but it but it looks like that. What can you do? Well, what we'd have to do is decide which one is smaller a or b. So if b is smaller than a, then b over a is smaller than 1, and we can multiply top and bottom by 1 over a. If you do the algebra there, then what you end up with is being that f is equal to 1 over a, which you don't really care about for the expansion, because this part here now looks like 1 over 1 minus x, that b over a is would be x. In which case the expansion will look like that's that 1 over a part, and here's the rest of it doing 1 plus x plus x squared and so forth. On the other hand, if we go back to this and we think that a is smaller than b, then what I need to do is multiply top and bottom by minus b. So I end up with that minus b in front, and then I have 1 over 1 minus x again. But now x is a over b, and so our convergent Taylor expansion looks like 1 plus a over b plus a over b squared and so on. And so that's the general idea behind Taylor expansions of functions of the form 1 over 1 minus x.