Well let's review a little bit of everything that we've covered so far. So what was the Maclaurin series representation of e to the x? And once again you’ll have to take my word for it that the Maclaurin series representation really does equal when you take the infinite series, it really does equal e to the x and I mentioned in the previous video that I was thinking about the proof and I finally gave up because I couldn’t think off the proof. And then I looked it up and then I realized why I couldn’t think of the proof, it's quite involve but I will do it eventually probably after I cover a lot of other things just because it's not something that you're really have to know to succeed well in calculus or appreciate what were about to do. But I will do it; it’ll probably five or six minutes.
Anyway, back to what were where we were. So the Maclaurin series representation of e to the x and it actually does equal e to the x, is one plus x plus x squared over two factorial, plus x to the third over three factorial. I'm going to do a bunch of terms and you'll see why, x to the fourth over four factorial, plus x to the five over five factorial, plus x to the sixth over six factorial, plus x to the seventh over seven factorial, plus x to the eighth over eight factorial and just this is a factorial. And it just keeps going on and on to infinity. Only, we took the infinite series does that exactly equal e to the x, fair enough.
Well what was the Maclaurin series representation of cosine of x? Cosine of x, well that equaled and I'm going to space them out a little, in a certain way and I think you’ll see why. It equaled one plus x squared over—oh, no sorry, one minus, this is a minus sign. It equal one minus x squared over two factorial, we’ve done that in the two videos ago; plus x to the fourth over four factorial; minus x to the sixth over six factorial. I think you might already know where I'm going with this. Plus x to the eighth over eight factorial and it just went and kept going in the next digit would just be a minus out here and it goes into infinity right for that pattern you know the pattern.
And what's the sine of x, what's the Maclaurin series representation for sine of x? Well sine of x it equals x minus x to the third over three factorial, plus x to the fifth over five factorial, minus x to the seventh over seven factorial and it just keeps going you’d have a x to the ninth out here but it goes on for infinity. This all go on for infinity but you know the pattern. So let's just pause here because this is, I mean I think if you understand what's going on one of the few things in life that will truly give you chills, it will truly make you believe that there is an order to the universe that we as human beings can only take, catch a glimpse off. We have limited minds but we are on the veered, we’re scrapping the surface here but there's something amazing here cosine of x and sine of x.
Each of their Maclaurin series representation or if you're essentially to write them a polynomials, each of them looks like it's almost like part of either the x. It's almost every other digit and they would be the same except for a couple of sine changes. Now, let me make that clear. So what I were to define the function cosine of x plus sine of x, cosine of x plus sine of x, what would that equal? Or what would its Maclaurin series representation of that be? Well and we know that it’s also equal but it's just essentially adding this two rows. So it would be 1 plus x minus x squared over two factorial, minus x to the third over three factorial, plus x to the fourth over four factorial, plus x to the fifth over five factorial, minus x to the sixth over six factorial, minus x to the seventh over seven factorial, plus x to the eighth over eight factorial and it just keeps going and the next one would be a plus but it goes on to infinity.
Now, I think it should be clear to you that something that the Goosebumps should be emerging on your arm. Because look at this and look at e to the x, what's the difference? Well it's just a couple of negative signs here and there. We have this negative sign, so the only difference between this function and this function are this negative signs. And I've seen this before, I've learned this before, people for hundreds of years have known this. But I’ll tell you something, no one even though they can prove it mathematically; no one really understands why this is. Why we take this, this trigonometric functions that appear when we take the ratios of the sides of a right triangle or the unit circle definition and it's useful for triangle measure, that’s trigonometry. That’s what we came up with this cosine and sine functions. And it's related to the circle and all of the rest.
That when you add this two fundamental functions together from trigonometry because tangent is really just the ratio of the sine and cosine. So these are really what trigonometry is, it is the basis of trigonometry. When you add their polynomial representations together it's almost exactly, almost exactly except for these negative signs. The polynomial representation of e to the x and e, and the number e—and first of all this is an exponent, these are no exponents here. And e it's completely unrelated or at least you one would think from trigonometry.
E is, we got e from compound interest; you’ll see that it's really to exponential growth, exponential decay, when you have continuous exponential growth and decay, continuous compound interest. It’s this number that is in a completely unrelated field of not just, but really the universe; continuous interest versus the ratios of the size of a right triangle. So this should already be getting you thinking. But what would be even more amazing is as if we could somehow work with this to make worth a little bit more equal. Well the only thing that’s the differences is this negative signs. So do we know anything else in mathematics, in our mathematical tool kit that has this pattern where it goes positive-positive-negative-negative-positive-positive-negative-negative and it has this essentially the cycle of four?
Well you might be thinking that this will even give you larger Goosebumps or make your current ones bigger than number i or the unit, imaginary unit i. So what are the powers of i? This is a little review, if this is completely unfamiliar to you, you should rewatch the imaginary numbers video. So what are the powers of I, well i to the zero is one, i to the first is i, i squared is negative one, i to the third power that’s negative one times i, so it's negative i; i to the fourth power is i times negative i, so the i has become negative one and then you have a negative there so it becomes one. And the pattern repeats itself; i to the fifth is i, i to the sixths is negative one, we learned this before but it's just review. I to the seventh is negative i, i to the eight then becomes one again.
So there you have it, this is amazing. I has a property where every, that the second two in this cycle four a negative. We have a negative number here. This is not necessarily a negative number, so negative imaginary but we have that negative sign so it looks pretty similar. Then we have two positives, do we have a negative and a negative. And something else is interesting going on here. Wherever we see the imaginary number, whenever we see an i or a negative i, which terms do they correspond to? Well they correspond to the terms of sine of x; they correspond to that term, negative i correspond to that term; i correspond to that term, negative i correspond to that term.
It seems a little bit even more of a pattern. But anyway, I just realized I only have, 40, I start to, I mean I have to say I’m normally pretty smooth in these videos but when I start talking about what I'm talking about right now, my brain starts to go in circles because this is, I've actually even heard this what we’re about to touch on as proof of the existence of God. And really that’s not that much of an exaggeration. It is definitely proof of the existence of some hidden order of the universe that we can only catch a glimpse off. And maybe you can call that God. But anyway I don’t want to get metaphysical on you. But I will see you in the next video.
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