1 00:00:14,640 --> 00:00:17,394 When I first learned about Taylor series, I definitely 2 00:00:17,394 --> 00:00:19,699 didn't appreciate just how important they are. 3 00:00:20,120 --> 00:00:22,828 But time and time again they come up in math, physics, 4 00:00:22,827 --> 00:00:25,929 and many fields of engineering because they're one of the most 5 00:00:25,929 --> 00:00:29,179 powerful tools that math has to offer for approximating functions. 6 00:00:30,000 --> 00:00:32,710 I think one of the first times this clicked for me as a 7 00:00:32,710 --> 00:00:35,420 student was not in a calculus class but a physics class. 8 00:00:35,840 --> 00:00:40,103 We were studying a certain problem that had to do with the potential energy of a 9 00:00:40,103 --> 00:00:44,155 pendulum, and for that you need an expression for how high the weight of the 10 00:00:44,155 --> 00:00:48,472 pendulum is above its lowest point, and when you work that out it comes out to be 11 00:00:48,472 --> 00:00:53,000 proportional to 1 minus the cosine of the angle between the pendulum and the vertical. 12 00:00:53,579 --> 00:00:57,875 The specifics of the problem we were trying to solve are beyond the point here, 13 00:00:57,875 --> 00:01:02,493 but what I'll say is that this cosine function made the problem awkward and unwieldy, 14 00:01:02,493 --> 00:01:06,519 and made it less clear how pendulums relate to other oscillating phenomena. 15 00:01:07,459 --> 00:01:12,516 But if you approximate cosine of theta as 1 minus theta squared over 2, 16 00:01:12,516 --> 00:01:15,959 everything just fell into place much more easily. 17 00:01:16,659 --> 00:01:19,213 If you've never seen anything like this before, 18 00:01:19,213 --> 00:01:22,780 an approximation like that might seem completely out of left field. 19 00:01:23,819 --> 00:01:28,803 If you graph cosine of theta along with this function, 1 minus theta squared over 2, 20 00:01:28,804 --> 00:01:33,203 they do seem rather close to each other, at least for small angles near 0, 21 00:01:33,203 --> 00:01:36,546 but how would you even think to make this approximation, 22 00:01:36,546 --> 00:01:39,420 and how would you find that particular quadratic? 23 00:01:41,219 --> 00:01:44,864 The study of Taylor series is largely about taking non-polynomial 24 00:01:44,864 --> 00:01:48,840 functions and finding polynomials that approximate them near some input. 25 00:01:48,840 --> 00:01:52,403 The motive here is that polynomials tend to be much easier to deal 26 00:01:52,403 --> 00:01:55,277 with than other functions, they're easier to compute, 27 00:01:55,277 --> 00:01:59,480 easier to take derivatives, easier to integrate, just all around more friendly. 28 00:02:00,680 --> 00:02:03,819 So let's take a look at that function, cosine of x, 29 00:02:03,819 --> 00:02:08,408 and really take a moment to think about how you might construct a quadratic 30 00:02:08,407 --> 00:02:10,219 approximation near x equals 0. 31 00:02:10,939 --> 00:02:16,444 That is, among all of the possible polynomials that look like c0 plus c1 32 00:02:16,444 --> 00:02:21,949 times x plus c2 times x squared, for some choice of these constants, c0, 33 00:02:21,949 --> 00:02:27,530 c1, and c2, find the one that most resembles cosine of x near x equals 0, 34 00:02:27,531 --> 00:02:32,659 whose graph kind of spoons with the graph of cosine x at that point. 35 00:02:33,860 --> 00:02:38,218 Well, first of all, at the input 0, the value of cosine of x is 1, 36 00:02:38,217 --> 00:02:41,861 so if our approximation is going to be any good at all, 37 00:02:41,861 --> 00:02:44,919 it should also equal 1 at the input x equals 0. 38 00:02:45,819 --> 00:02:50,939 Plugging in 0 just results in whatever c0 is, so we can set that equal to 1. 39 00:02:53,080 --> 00:02:56,608 This leaves us free to choose constants c1 and c2 to make this 40 00:02:56,608 --> 00:03:00,192 approximation as good as we can, but nothing we do with them is 41 00:03:00,192 --> 00:03:04,000 going to change the fact that the polynomial equals 1 at x equals 0. 42 00:03:04,960 --> 00:03:08,153 It would also be good if our approximation had the same 43 00:03:08,153 --> 00:03:11,120 tangent slope as cosine x at this point of interest. 44 00:03:11,900 --> 00:03:14,484 Otherwise the approximation drifts away from the 45 00:03:14,484 --> 00:03:16,700 cosine graph much faster than it needs to. 46 00:03:18,199 --> 00:03:22,146 The derivative of cosine is negative sine, and at x equals 0, 47 00:03:22,146 --> 00:03:25,840 that equals 0, meaning the tangent line is perfectly flat. 48 00:03:26,960 --> 00:03:31,920 On the other hand, when you work out the derivative of our quadratic, 49 00:03:31,919 --> 00:03:34,399 you get c1 plus 2 times c2 times x. 50 00:03:35,319 --> 00:03:39,419 At x equals 0, this just equals whatever we choose for c1. 51 00:03:40,259 --> 00:03:43,299 So this constant c1 has complete control over the 52 00:03:43,300 --> 00:03:46,340 derivative of our approximation around x equals 0. 53 00:03:47,120 --> 00:03:49,979 Setting it equal to 0 ensures that our approximation 54 00:03:49,979 --> 00:03:52,300 also has a flat tangent line at this point. 55 00:03:53,000 --> 00:03:57,810 This leaves us free to change c2, but the value and the slope of our 56 00:03:57,810 --> 00:04:02,620 polynomial at x equals 0 are locked in place to match that of cosine. 57 00:04:04,259 --> 00:04:08,378 The final thing to take advantage of is the fact that the cosine graph 58 00:04:08,378 --> 00:04:12,439 curves downward above x equals 0, it has a negative second derivative. 59 00:04:13,379 --> 00:04:17,250 Or in other words, even though the rate of change is 0 at that point, 60 00:04:17,250 --> 00:04:20,459 the rate of change itself is decreasing around that point. 61 00:04:21,278 --> 00:04:25,445 Specifically, since its derivative is negative sine of x, 62 00:04:25,446 --> 00:04:31,840 its second derivative is negative cosine of x, and at x equals 0, that equals negative 1. 63 00:04:33,079 --> 00:04:37,149 Now in the same way that we wanted the derivative of our approximation to 64 00:04:37,149 --> 00:04:41,933 match that of the cosine so that their values wouldn't drift apart needlessly quickly, 65 00:04:41,934 --> 00:04:45,784 making sure that their second derivatives match will ensure that they 66 00:04:45,783 --> 00:04:49,689 curve at the same rate, that the slope of our polynomial doesn't drift 67 00:04:49,689 --> 00:04:53,319 away from the slope of cosine x any more quickly than it needs to. 68 00:04:54,220 --> 00:04:59,225 Pulling up the same derivative we had before, and then taking its derivative, 69 00:04:59,225 --> 00:05:04,040 we see that the second derivative of this polynomial is exactly 2 times c2. 70 00:05:04,959 --> 00:05:10,435 So to make sure that this second derivative also equals negative 1 at x equals 0, 71 00:05:10,435 --> 00:05:15,579 2 times c2 has to be negative 1, meaning c2 itself should be negative 1 half. 72 00:05:16,379 --> 00:05:22,139 This gives us the approximation 1 plus 0x minus 1 half x squared. 73 00:05:23,199 --> 00:05:29,976 To get a feel for how good it is, if you estimate cosine of 0.1 using this polynomial, 74 00:05:29,976 --> 00:05:35,819 you'd estimate it to be 0.995, and this is the true value of cosine of 0.1. 75 00:05:36,639 --> 00:05:38,439 It's a really good approximation! 76 00:05:40,300 --> 00:05:42,520 Take a moment to reflect on what just happened. 77 00:05:42,519 --> 00:05:46,992 You had 3 degrees of freedom with this quadratic approximation, 78 00:05:46,992 --> 00:05:49,019 the constants c0, c1, and c2. 79 00:05:49,519 --> 00:05:55,806 c0 was responsible for making sure that the output of the approximation matches that of 80 00:05:55,807 --> 00:06:01,951 cosine x at x equals 0, c1 was in charge of making sure that the derivatives match at 81 00:06:01,951 --> 00:06:08,240 that point, and c2 was responsible for making sure that the second derivatives match up. 82 00:06:08,939 --> 00:06:14,271 This ensures that the way your approximation changes as you move away from x equals 0, 83 00:06:14,271 --> 00:06:17,459 and the way that the rate of change itself changes, 84 00:06:17,459 --> 00:06:20,891 is as similar as possible to the behaviour of cosine x, 85 00:06:20,891 --> 00:06:23,159 given the amount of control you have. 86 00:06:24,079 --> 00:06:27,187 You could give yourself more control by allowing more terms 87 00:06:27,187 --> 00:06:30,139 in your polynomial and matching higher order derivatives. 88 00:06:30,839 --> 00:06:36,579 For example, let's say you added on the term c3 times x cubed for some constant c3. 89 00:06:36,579 --> 00:06:41,479 In that case, if you take the third derivative of a cubic polynomial, 90 00:06:41,480 --> 00:06:44,280 anything quadratic or smaller goes to 0. 91 00:06:45,560 --> 00:06:50,882 As for that last term, after 3 iterations of the power rule, 92 00:06:50,882 --> 00:06:54,459 it looks like 1 times 2 times 3 times c3. 93 00:06:56,459 --> 00:07:01,339 On the other hand, the third derivative of cosine x comes out to sine x, 94 00:07:01,339 --> 00:07:03,279 which equals 0 at x equals 0. 95 00:07:03,279 --> 00:07:08,759 So to make sure that the third derivatives match, the constant c3 should be 0. 96 00:07:09,879 --> 00:07:14,695 Or in other words, not only is 1 minus ½ x2 the best possible quadratic 97 00:07:14,696 --> 00:07:19,580 approximation of cosine, it's also the best possible cubic approximation. 98 00:07:21,279 --> 00:07:27,059 You can make an improvement by adding on a fourth order term, c4 times x to the fourth. 99 00:07:27,879 --> 00:07:33,319 The fourth derivative of cosine is itself, which equals 1 at x equals 0. 100 00:07:34,300 --> 00:07:37,460 And what's the fourth derivative of our polynomial with this new term? 101 00:07:38,620 --> 00:07:42,677 Well, when you keep applying the power rule over and over, 102 00:07:42,677 --> 00:07:45,979 with those exponents all hopping down in front, 103 00:07:45,978 --> 00:07:51,000 you end up with 1 times 2 times 3 times 4 times c4, which is 24 times c4. 104 00:07:51,399 --> 00:07:56,130 So if we want this to match the fourth derivative of cosine x, 105 00:07:56,130 --> 00:07:58,759 which is 1, c4 has to be 1 over 24. 106 00:07:59,819 --> 00:08:05,872 And indeed, the polynomial 1 minus ½ x2 plus 1 24 times x to the fourth, 107 00:08:05,872 --> 00:08:12,839 which looks like this, is a very close approximation for cosine x around x equals 0. 108 00:08:13,740 --> 00:08:18,112 In any physics problem involving the cosine of a small angle, for example, 109 00:08:18,112 --> 00:08:23,127 predictions would be almost unnoticeably different if you substituted this polynomial 110 00:08:23,127 --> 00:08:24,060 for cosine of x. 111 00:08:26,100 --> 00:08:29,760 Take a step back and notice a few things happening with this process. 112 00:08:30,519 --> 00:08:34,199 First of all, factorial terms come up very naturally in this process. 113 00:08:35,019 --> 00:08:39,800 When you take n successive derivatives of the function x to the n, 114 00:08:39,801 --> 00:08:43,156 letting the power rule keep cascading on down, 115 00:08:43,155 --> 00:08:48,579 what you'll be left with is 1 times 2 times 3 on and on up to whatever n is. 116 00:08:49,220 --> 00:08:53,988 So you don't simply set the coefficients of the polynomial equal to whatever derivative 117 00:08:53,988 --> 00:08:58,540 you want, you have to divide by the appropriate factorial to cancel out this effect. 118 00:08:59,399 --> 00:09:05,343 For example, that x to the fourth coefficient was the fourth derivative of cosine, 119 00:09:05,344 --> 00:09:07,780 1, but divided by 4 factorial, 24. 120 00:09:09,399 --> 00:09:12,739 The second thing to notice is that adding on new terms, 121 00:09:12,739 --> 00:09:17,629 like this c4 times x to the fourth, doesn't mess up what the old terms should be, 122 00:09:17,629 --> 00:09:19,299 and that's really important. 123 00:09:20,100 --> 00:09:25,213 For example, the second derivative of this polynomial at x equals 0 is still equal 124 00:09:25,212 --> 00:09:30,079 to 2 times the second coefficient, even after you introduce higher order terms. 125 00:09:30,960 --> 00:09:33,879 And it's because we're plugging in x equals 0, 126 00:09:33,879 --> 00:09:38,537 so the second derivative of any higher order term, which all include an x, 127 00:09:38,537 --> 00:09:39,780 will just wash away. 128 00:09:40,740 --> 00:09:45,479 And the same goes for any other derivative, which is why each derivative of a 129 00:09:45,479 --> 00:09:50,280 polynomial at x equals 0 is controlled by one and only one of the coefficients. 130 00:09:52,639 --> 00:09:57,352 If instead you were approximating near an input other than 0, like x equals pi, 131 00:09:57,352 --> 00:10:01,771 in order to get the same effect you would have to write your polynomial in 132 00:10:01,772 --> 00:10:05,720 terms of powers of x minus pi, or whatever input you're looking at. 133 00:10:06,320 --> 00:10:09,208 This makes it look noticeably more complicated, 134 00:10:09,207 --> 00:10:13,961 but all we're doing is making sure that the point pi looks and behaves like 0, 135 00:10:13,961 --> 00:10:18,715 so that plugging in x equals pi will result in a lot of nice cancellation that 136 00:10:18,715 --> 00:10:20,220 leaves only one constant. 137 00:10:22,379 --> 00:10:27,730 And finally, on a more philosophical level, notice how what we're doing here is basically 138 00:10:27,730 --> 00:10:32,665 taking information about higher order derivatives of a function at a single point, 139 00:10:32,666 --> 00:10:37,780 and translating that into information about the value of the function near that point. 140 00:10:40,960 --> 00:10:44,120 You can take as many derivatives of cosine as you want. 141 00:10:44,600 --> 00:10:47,543 It follows this nice cyclic pattern, cosine of x, 142 00:10:47,543 --> 00:10:51,019 negative sine of x, negative cosine, sine, and then repeat. 143 00:10:52,320 --> 00:10:55,660 And the value of each one of these is easy to compute at x equals 0. 144 00:10:56,100 --> 00:11:01,100 It gives this cyclic pattern 1, 0, negative 1, 0, and then repeat. 145 00:11:02,000 --> 00:11:07,149 And knowing the values of all those higher order derivatives is a lot of information 146 00:11:07,149 --> 00:11:12,480 about cosine of x, even though it only involves plugging in a single number, x equals 0. 147 00:11:14,259 --> 00:11:19,602 So what we're doing is leveraging that information to get an approximation around this 148 00:11:19,602 --> 00:11:25,130 input, and you do it by creating a polynomial whose higher order derivatives are designed 149 00:11:25,130 --> 00:11:30,659 to match up with those of cosine, following this same 1, 0, negative 1, 0, cyclic pattern. 150 00:11:31,419 --> 00:11:35,481 And to do that, you just make each coefficient of the polynomial follow that 151 00:11:35,481 --> 00:11:39,439 same pattern, but you have to divide each one by the appropriate factorial. 152 00:11:40,120 --> 00:11:42,615 Like I mentioned before, this is what cancels out 153 00:11:42,615 --> 00:11:45,259 the cascading effect of many power rule applications. 154 00:11:47,279 --> 00:11:50,110 The polynomials you get by stopping this process at 155 00:11:50,110 --> 00:11:53,159 any point are called Taylor polynomials for cosine of x. 156 00:11:53,899 --> 00:11:58,659 More generally, and hence more abstractly, if we were dealing with some other function 157 00:11:58,659 --> 00:12:03,419 other than cosine, you would compute its derivative, its second derivative, and so on, 158 00:12:03,419 --> 00:12:08,289 getting as many terms as you'd like, and you would evaluate each one of them at x equals 159 00:12:08,289 --> 00:12:08,399 0. 160 00:12:09,580 --> 00:12:15,937 Then for the polynomial approximation, the coefficient of each x to the n term should be 161 00:12:15,937 --> 00:12:20,510 the value of the nth derivative of the function evaluated at 0, 162 00:12:20,510 --> 00:12:22,439 but divided by n factorial. 163 00:12:23,480 --> 00:12:27,370 This whole rather abstract formula is something you'll likely 164 00:12:27,370 --> 00:12:31,199 see in any text or course that touches on Taylor polynomials. 165 00:12:31,779 --> 00:12:36,139 And when you see it, think to yourself that the constant term ensures that 166 00:12:36,139 --> 00:12:39,451 the value of the polynomial matches with the value of f, 167 00:12:39,451 --> 00:12:43,696 the next term ensures that the slope of the polynomial matches the slope 168 00:12:43,696 --> 00:12:48,114 of the function at x equals 0, the next term ensures that the rate at which 169 00:12:48,114 --> 00:12:51,369 the slope changes is the same at that point, and so on, 170 00:12:51,369 --> 00:12:53,519 depending on how many terms you want. 171 00:12:54,620 --> 00:12:57,451 And the more terms you choose, the closer the approximation, 172 00:12:57,451 --> 00:13:00,980 but the tradeoff is that the polynomial you'd get would be more complicated. 173 00:13:02,639 --> 00:13:07,726 And to make things even more general, if you wanted to approximate near some input 174 00:13:07,726 --> 00:13:12,937 other than 0, which we'll call a, you would write this polynomial in terms of powers 175 00:13:12,937 --> 00:13:17,779 of x minus a, and you would evaluate all the derivatives of f at that input, a. 176 00:13:18,679 --> 00:13:23,120 This is what Taylor polynomials look like in their fullest generality. 177 00:13:24,000 --> 00:13:28,533 Changing the value of a changes where this approximation is hugging the original 178 00:13:28,533 --> 00:13:33,235 function, where its higher order derivatives will be equal to those of the original 179 00:13:33,235 --> 00:13:33,740 function. 180 00:13:35,879 --> 00:13:38,860 One of the simplest meaningful examples of this is 181 00:13:38,860 --> 00:13:41,899 the function e to the x around the input x equals 0. 182 00:13:42,759 --> 00:13:46,405 Computing the derivatives is super nice, as nice as it gets, 183 00:13:46,405 --> 00:13:49,274 because the derivative of e to the x is itself, 184 00:13:49,274 --> 00:13:53,579 so the second derivative is also e to the x, as is its third, and so on. 185 00:13:54,340 --> 00:13:58,240 So at the point x equals 0, all of these are equal to 1. 186 00:13:59,120 --> 00:14:05,720 And what that means is our polynomial approximation should look like 187 00:14:05,720 --> 00:14:13,947 1 plus 1 times x plus 1 over 2 times x squared plus 1 over 3 factorial times x cubed, 188 00:14:13,947 --> 00:14:18,539 and so on, depending on how many terms you want. 189 00:14:19,399 --> 00:14:22,699 These are the Taylor polynomials for e to the x. 190 00:14:26,379 --> 00:14:31,038 Ok, so with that as a foundation, in the spirit of showing you just how connected all 191 00:14:31,038 --> 00:14:34,614 the topics of calculus are, let me turn to something kind of fun, 192 00:14:34,614 --> 00:14:38,840 a completely different way to understand this second order term of the Taylor 193 00:14:38,840 --> 00:14:40,519 polynomials, but geometrically. 194 00:14:41,399 --> 00:14:43,903 It's related to the fundamental theorem of calculus, 195 00:14:43,903 --> 00:14:47,259 which I talked about in chapters 1 and 8 if you need a quick refresher. 196 00:14:47,980 --> 00:14:52,000 Like we did in those videos, consider a function that gives the area 197 00:14:52,000 --> 00:14:56,139 under some graph between a fixed left point and a variable right point. 198 00:14:56,980 --> 00:15:00,914 What we're going to do here is think about how to approximate this area function, 199 00:15:00,914 --> 00:15:04,179 not the function for the graph itself, like we've been doing before. 200 00:15:04,899 --> 00:15:09,439 Focusing on that area is what's going to make the second order term pop out. 201 00:15:10,440 --> 00:15:16,575 Remember, the fundamental theorem of calculus is that this graph itself represents the 202 00:15:16,575 --> 00:15:22,711 derivative of the area function, and it's because a slight nudge dx to the right bound 203 00:15:22,711 --> 00:15:28,988 of the area gives a new bit of area approximately equal to the height of the graph times 204 00:15:28,988 --> 00:15:29,200 dx. 205 00:15:30,039 --> 00:15:34,480 And that approximation is increasingly accurate for smaller and smaller choices of dx. 206 00:15:35,980 --> 00:15:39,600 But if you wanted to be more accurate about this change in area, 207 00:15:39,600 --> 00:15:42,666 given some change in x that isn't meant to approach 0, 208 00:15:42,666 --> 00:15:46,065 you would have to take into account this portion right here, 209 00:15:46,065 --> 00:15:47,960 which is approximately a triangle. 210 00:15:49,600 --> 00:15:57,460 Let's name the starting input a, and the nudged input above it x, so that change is x-a. 211 00:15:58,100 --> 00:16:02,985 The base of that little triangle is that change, x-a, 212 00:16:02,985 --> 00:16:07,600 and its height is the slope of the graph times x-a. 213 00:16:08,419 --> 00:16:11,986 Since this graph is the derivative of the area function, 214 00:16:11,986 --> 00:16:17,120 its slope is the second derivative of the area function, evaluated at the input a. 215 00:16:18,440 --> 00:16:22,662 So the area of this triangle, 1 half base times height, 216 00:16:22,662 --> 00:16:28,466 is 1 half times the second derivative of this area function, evaluated at a, 217 00:16:28,466 --> 00:16:29,899 multiplied by x-a2. 218 00:16:30,960 --> 00:16:34,379 And this is exactly what you would see with a Taylor polynomial. 219 00:16:34,879 --> 00:16:40,477 If you knew the various derivative information about this area function at the point a, 220 00:16:40,477 --> 00:16:43,659 how would you approximate the area at the point x? 221 00:16:45,360 --> 00:16:49,316 Well you have to include all that area up to a, f of a, 222 00:16:49,316 --> 00:16:54,968 plus the area of this rectangle here, which is the first derivative, times x-a, 223 00:16:54,967 --> 00:17:00,901 plus the area of that little triangle, which is 1 half times the second derivative, 224 00:17:00,902 --> 00:17:01,680 times x-a2. 225 00:17:02,559 --> 00:17:06,538 I really like this, because even though it looks a bit messy all written out, 226 00:17:06,538 --> 00:17:11,078 each one of the terms has a very clear meaning that you can just point to on the diagram. 227 00:17:13,400 --> 00:17:16,877 If you wanted, we could call it an end here, and you would have a 228 00:17:16,876 --> 00:17:20,460 phenomenally useful tool for approximating these Taylor polynomials. 229 00:17:21,400 --> 00:17:25,812 But if you're thinking like a mathematician, one question you might ask is 230 00:17:25,811 --> 00:17:30,459 whether or not it makes sense to never stop and just add infinitely many terms. 231 00:17:31,380 --> 00:17:35,883 In math, an infinite sum is called a series, so even though one of these 232 00:17:35,883 --> 00:17:40,263 approximations with finitely many terms is called a Taylor polynomial, 233 00:17:40,262 --> 00:17:44,519 adding all infinitely many terms gives what's called a Taylor series. 234 00:17:45,259 --> 00:17:48,847 You have to be really careful with the idea of an infinite series, 235 00:17:48,847 --> 00:17:52,597 because it doesn't actually make sense to add infinitely many things, 236 00:17:52,597 --> 00:17:56,079 you can only hit the plus button on the calculator so many times. 237 00:17:57,440 --> 00:18:01,380 But if you have a series where adding more and more of the terms, 238 00:18:01,380 --> 00:18:06,396 which makes sense at each step, gets you increasingly close to some specific value, 239 00:18:06,395 --> 00:18:09,740 what you say is that the series converges to that value. 240 00:18:10,319 --> 00:18:14,292 Or, if you're comfortable extending the definition of equality to 241 00:18:14,292 --> 00:18:19,048 include this kind of series convergence, you'd say that the series as a whole, 242 00:18:19,048 --> 00:18:22,359 this infinite sum, equals the value it's converging to. 243 00:18:23,460 --> 00:18:27,451 For example, look at the Taylor polynomial for e to the x, 244 00:18:27,451 --> 00:18:30,160 and plug in some input, like x equals 1. 245 00:18:31,140 --> 00:18:36,278 As you add more and more polynomial terms, the total sum gets closer and 246 00:18:36,278 --> 00:18:42,404 closer to the value e, so you say that this infinite series converges to the number e, 247 00:18:42,404 --> 00:18:46,700 or what's saying the same thing, that it equals the number e. 248 00:18:47,839 --> 00:18:53,638 In fact, it turns out that if you plug in any other value of x, like x equals 2, 249 00:18:53,638 --> 00:18:59,866 and look at the value of the higher and higher order Taylor polynomials at this value, 250 00:18:59,866 --> 00:19:04,019 they will converge towards e to the x, which is e squared. 251 00:19:04,680 --> 00:19:08,951 This is true for any input, no matter how far away from 0 it is, 252 00:19:08,951 --> 00:19:14,602 even though these Taylor polynomials are constructed only from derivative information 253 00:19:14,602 --> 00:19:16,180 gathered at the input 0. 254 00:19:18,269 --> 00:19:24,276 In a case like this, we say that e to the x equals its own Taylor series at all inputs x, 255 00:19:24,276 --> 00:19:27,480 which is kind of a magical thing to have happen. 256 00:19:28,380 --> 00:19:32,400 Even though this is also true for a couple other important functions, 257 00:19:32,400 --> 00:19:36,306 like sine and cosine, sometimes these series only converge within a 258 00:19:36,306 --> 00:19:40,500 certain range around the input whose derivative information you're using. 259 00:19:41,579 --> 00:19:47,272 If you work out the Taylor series for the natural log of x around the input x equals 1, 260 00:19:47,272 --> 00:19:51,995 which is built by evaluating the higher order derivatives of the natural 261 00:19:51,996 --> 00:19:55,620 log of x at x equals 1, this is what it would look like. 262 00:19:56,079 --> 00:20:00,799 When you plug in an input between 0 and 2, adding more and more terms of this 263 00:20:00,799 --> 00:20:05,519 series will indeed get you closer and closer to the natural log of that input. 264 00:20:06,400 --> 00:20:09,509 But outside of that range, even by just a little bit, 265 00:20:09,509 --> 00:20:11,700 the series fails to approach anything. 266 00:20:12,480 --> 00:20:17,440 As you add on more and more terms, the sum bounces back and forth wildly. 267 00:20:18,099 --> 00:20:22,694 It does not, as you might expect, approach the natural log of that value, 268 00:20:22,694 --> 00:20:27,539 even though the natural log of x is perfectly well defined for inputs above 2. 269 00:20:28,460 --> 00:20:31,838 In some sense, the derivative information of ln 270 00:20:31,838 --> 00:20:35,359 of x at x equals 1 doesn't propagate out that far. 271 00:20:36,579 --> 00:20:41,276 In a case like this, where adding more terms of the series doesn't approach anything, 272 00:20:41,277 --> 00:20:43,080 you say that the series diverges. 273 00:20:44,180 --> 00:20:47,952 And that maximum distance between the input you're approximating 274 00:20:47,952 --> 00:20:51,668 near and points where the outputs of these polynomials actually 275 00:20:51,669 --> 00:20:55,560 converge is called the radius of convergence for the Taylor series. 276 00:20:56,839 --> 00:20:59,159 There remains more to learn about Taylor series. 277 00:20:59,500 --> 00:21:03,241 There are many use cases, tactics for placing bounds on the error of 278 00:21:03,241 --> 00:21:07,635 these approximations, tests for understanding when series do and don't converge, 279 00:21:07,635 --> 00:21:11,759 and for that matter, there remains more to learn about calculus as a whole, 280 00:21:11,759 --> 00:21:14,579 and the countless topics not touched by this series. 281 00:21:15,319 --> 00:21:19,240 The goal with these videos is to give you the fundamental intuitions 282 00:21:19,240 --> 00:21:23,388 that make you feel confident and efficient in learning more on your own, 283 00:21:23,388 --> 00:21:27,139 and potentially even rediscovering more of the topic for yourself. 284 00:21:28,059 --> 00:21:32,326 In the case of Taylor series, the fundamental intuition to keep in mind 285 00:21:32,326 --> 00:21:36,595 as you explore more of what there is, is that they translate derivative 286 00:21:36,595 --> 00:21:41,160 information at a single point to approximation information around that point. 287 00:21:43,920 --> 00:21:46,600 Thank you once again to everybody who supported this series. 288 00:21:47,299 --> 00:21:49,528 The next series like it will be on probability, 289 00:21:49,528 --> 00:21:53,059 and if you want early access as those videos are made, you know where to go. 290 00:22:11,160 --> 00:22:19,060 Thank you.