Worked example: cosine function from power series

- So I have an infinite series, here. The sum from n equals zero to infinity of negative one to the n power times x to the sixth n over two n, the whole two n, factorial. And my goal in this video is to evaluate this power series when x is equal to the cubed root of pi over two. And I encourage you to pause this video and give it a go on your own. And I will give you a hint. The key to this is to figure out, well what function is this the power series for? and then use that function to evaluate this. And there's another clue here, hey, this is kind of a mysterious or a suspicious looking number, here, pi over two. That looks like something I would use a trig function to evaluate. That might be a little bit more straightforward. So I'll let you have a go at it. So I'm assuming you have tried, so let's try to work through this together. And in any of these types of problems I like to at least expand out this power series so I get a better sense of what it's like. So this right over here, if I were to expand it out, this is going to be equal to when n is zero, this is one, actually, all of these are one, so it's just going to be one. When n is one, this is going to be negative one, x to the sixth, x to the sixth over two factorial. when n is two, it's going to be positive. negative one squared is positive one, times x to the 12th over four factorial. and then let's just do one more: when x is equal to three, it's going to be negative x to the 18th. x to the 18th over six factorial. And you just keep going on and on forever. Now, offhand, I don't know a function, especially a trigonometric function, because that was kind of our clue here, this pi over two makes me feel like this might be a trigonometric function right over here Nothing jumps out at me offhand, but this does look suspiciously familiar. This looks awfully close to the power series, or the Maclaurin Series for cosine of x. Which we have seen multiple times. Let's just remind ourselves of what that is. And if this doesn't look familiar, the previous video where I do the Maclaurin Series for cosine of x goes into detail on how I get this. The Maclaurin Series for cosine of x is equal to... So I'll just write a few terms, so I'll write approximately equal to one minus x squared over two factorial, plus x to the fourth over four factorial, minus x to the sixth over six factorial, and just like that you're probably seeing the similarities. Well the first term is the same, the sign negative, positive, negative, positive, negative, positive, negative, positive. two factorial, four factorial, six factorial. The difference is the powers, the exponents on the x. This is x squared, this is x to the sixth. This is x to the fourth, that's x to the 12th. This is x to the sixth, that's x to the 18th. Well, what if we... I guess, something for you to think about is, well, how can we replace x with something here? Because anything that I change, if I take cosine of... if I change x to, I don't know, a plus b, everywhere we see an x, you'd replace it with an a plus b. Can we put a power of x here, so that these things end up like that? well, this x to the sixth is the same thing, as x to the third squared. That's x to the third squared. This right over here, x to the 12th, is the same thing as x to the third to the fourth power. This right over here is the same thing as x to the third to the sixth power. So if we could replace each of these xs with x to the thirds we will get this power series up here. Well, how do we do that? Well, we would just say the cosine of x to the third and actually let me do that in a different color. So the cosine, and that's not a different color. So the cosine of x to the third is going to be equal to, and once again, everywhere we see an x, we replace it with x to the third. So it's one minus, and actually, I'm just going to put in parenthesis squared, two factorial. I wanted to do that in the green, let me do all this in the green. Alright, so it's gonna be equal to: one minus parenthesis squared over two factorial plus parenthesis to the fourth power over four factorial. minus parenthesis to the sixth power over 6 factorial. And now let me change back to that mauve color. And since I'm taking the cosine of x to the third, well this is gonna be x to the third squared. this is gonna be x to the third to the fourth power. this is gonna be x to the third to the sixth power. Which is exactly what I have right over here. So this right over here is the power series for cosine of x to the third. So evaluating this when x is equal to the cubed root of pi over two, is the same thing as evaluating this when x is equal to the cubed root of pi over two. Let me write that down, because this is interesting. So this, so I'll just rewrite it, from n equals zero to infinity of negative one to the n of x to the sixth n over two n factorial. This is equal to, this is the power series representation, of cosine of x to the third power. So if you want to evaluate this when x is the cubed root of pi over two, we just have to evaluate this when x is the cubed root of pi over two. And this does suspiciously work out nicely, because if you take the cube of the cubed root, well, good things happen. So the cosine of the cubed root of pi over two to the third power, well that's just the same thing as cosine of pi over two, which, of course, is equal to zero. And we are done.