 In this video, we're going to discuss the solution of question six from the final exam for math 12-20. And in this question, we're asked to find the Maclaurin series for the function f of x equals x times e to the x. Well, we could try to, you know, just make this thing from scratch, but it is a multiple choice question. Probably there's a much simpler way than doing that. And so the idea here is that we know the Maclaurin series for e to the x. This is one of the most important Maclaurin series that we're told to memorize. Or at the very least, we put that, we put that formula on a note sheet that we can use on this test. In which case e to the x, we see that its Maclaurin series is the sum or n equals zero to infinity. We get x to the n over n factorial. And so to get the Maclaurin series for the function we want, we're going to times everything by x. But as this is a sum, we can distribute the x through, in which case this gives us the sum where n still goes from zero to infinity, that didn't change. And the power of x is going to increase by one, we get x to the n plus one over n factorial. And so we then would select choice f, which is the correct one there. And so you can see that, I mean, this one right here is just the Maclaurin series for e to the x. So you're so excited you found the right Maclaurin series for e to the x, you forget. You didn't adjust by x. This one right here is actually the Maclaurin series for sine. This one right here is the Maclaurin series for cosine. I should be exiting those because those are not correct. And then the other two are if you multiply sine. So this is the Maclaurin series for x times sine of x. And this last one is the Maclaurin series for what's happening there. It looks like x squared times cosine of x or something like that. And so neither one of those are correct. You didn't have to go through an analysis, but just kind of give you some idea where these things come from. You're going to want to know the Maclaurin series for e to the x, for sine of x, cosine of x. Natural log of one plus x, the geometric series, arc tangents, geometric series, the binomial series. Those are the important ones. You're going to want to memorize for an exam like this. Not just so you can answer this question type quickly, but it'll be useful in other questions as well.