I had dinner between the last video and this one so I might have forgotten what I just did. I think I was about to if what I see on my board make sense. I was about to use the tailor scissors or in this specific example the Maclaurin series approximation to figure out a polynomial version of a sum of polynomial terms to approximate ex. And remember, let me write here what the definition of the Maclaurin series was, it’s that we said that f(x) is equal to the sum from n = 0 to infinity of the nth derivative of f evaluated that zero. And if I remember, they put it a value than zero the last time I wrote this down, times x to the n over n factorial and hopefully that makes sense to you. This might seem really confusing and strange but now that we are going to apply it to ex, this should maybe be a little bit more concrete. And I think at the end of my last video, I said, “Well, if f(x) is ex, f(0) is e0 which is one and f’(x), I mean that’s any derivative of ex is equal to ex .
So, you take any derivative at zero added equals one for ex for this particular case of f(x) and that’s really neat. That means that the rate of change of y with respect to x is you know for every one you move in x, you move one in y, that’s fine. Add f at e0 but that also means that the rate of change of the rate of change is also one of the rate of change of the rate of change at the rate of change is also one so at e0 or x=0 of ex, the slope of the slope of the slope of the slope of the slope of the slope of the slope, they’re all one which to me tells me something is mysterious is happening. It’s another reason why you should just sit and ponder e.
But anyway, back to what we were trying to do. So, how would we do this? How would we write the approximation? Well, we could say that—let me write the approximate. I'll call that p(x) because it’s going to be a—it equals. Well, in this particular case, what’s the derivative of any derivative evaluated at f(0)? Well, that term is one. We wrote that down right here. f(0) is one, the derivative, the first derivative at zero is one, the second derivative at zero is one. That’s what special about the ex so all of these terms are going to equal one. So, this polynomial simplifies to the sum from n equals zero to infinity of x to the n over n factorial. That to me is very neat.
Remember, these are all one in every term so that’s why I took it out. So, what does that mean? Well, that tells us that ex can be approximated and actually I don’t prove it here but it actually turns out that we take the infinite sum that the Maclaurin series does not only approximates ex at x equals zero. When you take the infinite series, it actually equals ex. So, when you take a Maclaurin series at zero and the resulting function, the resulting polynomial actually converges and that’s something we will learn a little bit more rigorously hopefully later when we start doing analysis but it could actually converge to the function at all points and it actually is the case with ex. So, we can actually say that ex is equal—I didn’t prove this but you can take my word for and you can even test it out with some numbers. It equals this sum but what is the sum?
Well, it’s x0 over zero factorial so let me actually just write x0 over zero factorial plus x1 over one factorial plus x squared over two factorial. And you keep going and of course that’s equal so ex is equal to x0 is one, zero factorial I said in the last video is one so it’s one plus. This is just x plus x2 over two factorial plus x3 over three factorial plus x4 over four factorial and you just keep going on forever. And that’s ex and to me, that is amazing because this strange number e, this 2.7, you know whatever, whatever that we got from compound interest, it can be written as an infinite polynomial, this polynomial series or this Maclaurin series that actually has a certain beauty to it right? This number is kind of ugly when you write it out 2.7 whatever, whatever but when you write it to an exponent power as an infinite sum, it kind of has a nice rhythm to it. It is a very patterned and who would have got guest that you could have written it in such a simple form and even more. What happens when x is equal to one? So, what’s e1? Well then, we said x equal to one on both sides. And I think I have space to do it here, e1 which is equal to e, we just set all the axis to one so we get—that’s equal to one plus one plus one over two factorial plus one over three factorial plus one over four factorial plus one over five factorial.
That to me once again is amazing that the number e and we have just stumbled on another definition of e. E is equal to the sum from n equals zero to infinity of one over n factorial. That is amazing.
So now, we have two definitions for e. We have this one that we stumbled on and of course we had the ones from compound interest that I will do in magenta. The limit as n approaches infinity of one plus one over n to the n, that also is equal to e. This is starting to give me chills because this very strange bizarre number is popping out, kind of this might not seem so natural to you but it’s neat and it comes out in compound interest and continuously compounding interest. But this is even simpler. I just keep taking one over the factorial of numbers and I add them all together. And if I take every number really in existence and I sum them all up, I get e. That to me is amazing. One over n factorial of essentially every integer from zero to infinity, if I sum them up, I get the number e. You hopefully are getting chills right now.
Well anyway, let’s do the Maclaurin series for a couple of more functions and then we will get to something that is even more mind blowing. I'll see you in the next video.
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