 So, remember we found the Taylor series for f of x at x equals a by assuming a power series expansion for f of x, and repeatedly differentiating and evaluating at x equals a allows us to find the coefficients a0, a1, and so on. If we do this generically, we produce a formula for the coefficients. But as always, don't memorize formulas, understand concepts. So for example, suppose you want to find the nth coefficient of the Maclaurin series for some function. So remember the Maclaurin series is the Taylor series at x equals 0, and our concept is we're assuming that there is some power series that gives us f of x. Now if x equals 0, then we find that f of 0 is, which gives us a0. To find the other coefficients, let's differentiate term by term. And remember this is an infinite series. So the first term we didn't write down was a4x to the fourth. When we differentiate, it contributes a4, a4x cubed. And if x equals 0, we find that f prime of 0 is, and that gives us our a1 coefficient. If we differentiate again, we find, where again, because it's an infinite series, we get some additional terms that we didn't initially write down. And here we'll leave our coefficients in factored form, because it's useful in higher mathematics not to do arithmetic. And we find our second derivative at 0 is, and so our a2 coefficient must be. And now, lather, rinse, repeat, we'll find our derivative, we'll solve for our third derivative at 0, then solve for a3, and a4, and now we want to generalize. And here's a useful bit of notation. Notice that we're getting these products of decreasing whole numbers. So a useful bit of notation is that the product of the whole numbers, from 1 through n, is designated n factorial, where 0 factorial is defined to be 1. So this a4 term with denominator 4 times 3 times 2, we can rewrite 4 times 3 times 2 as 4 times 3 times 2 times 1, or 4 factorial. And so a4 is the fourth derivative divided by 4 factorial. Our third derivative divided by 3 times 2, well, 3 times 2 is really 3 times 2 times 1, or 3 factorial. Our a3 term is the third derivative divided by 3 factorial. 2 is 2 times 1, or 2 factorial. So a2 is our second derivative divided by 2 factorial. Now consistency is nice, and these first two terms are a little bit inconsistent. They're not divided by anything. Or are they? Remember we can rewrite anything as a fraction with denominator 1. And 1 is 1 factorial, or it's also 0 factorial. And that means we could write our first two terms in the same way as all of the others. And that suggests in general the nth coefficient will be the nth derivative at 0 divided by n factorial. And this leads to the following result, where if you're wise about it, you will ignore this theorem, because don't memorize formulas, understand concepts. The Taylor and McLaurin series come from assuming a power series expanded for our function and successively differentiating. But it's nice to have a formula to refer to at least, so we'll use the following, the Taylor series for a function at x equals a is given by this infinite series, where our coefficients are the nth derivatives divided by n factorial. Now there's about three times in all of mathematics where you'd want to find the Taylor or McLaurin series from the theorem. This is one of them. Let's find the McLaurin series for f of x equals e to the x, and then write down the first four terms. So remember the McLaurin series is the Taylor series at x equals 0. And we have our formula that says we need to find the nth derivative. The only reason that it's useful in this case is that e to the x and all of its derivatives are e to the x. And so this nth derivative has a nice formula, likewise the nth derivative at 0. And so our nth coefficient is going to be 1 divided by n factorial, which allows us to write down our Taylor series for e to the x in a nice summation notation. But if we actually want to use it, we can write out the first couple of terms. n equals 0, n equals 1, n equals 2, n equals 3. And there are more terms that we won't write down, and we'll do a little bit of algebra.