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    This is true for any Taylor expansion of any finite function… The exponential is pretty incredible, but not so much because its Taylor expansion behaves like every other Taylor expansion…

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      What do you mean by finite? Bounded? Exp isn’t bounded and yet its expansion at any point converges to e^x for all x. But e^(-1/x) (for x > 0, 0 otherwise) is both bounded and smooth yet it’s not analytic, i.e. T[f, 0](x) = 0 for all x.

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        Finite just means it never reaches infinity, whereas bounded means it has a real maximum value.

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          Silly me

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      Another cool thing about exponents: they are the most simple mathematical computation:

      (From my notes)

      // Exponents are just numbers applied to each other.
      // _3(_2) = 2^3
      // Which expands the _2 eight times. But it's easier to read when use it this way:
      const Exp = m => n => n(m);
      // Exp(2)(3) === 2^3 which is more in-line with what we're used to.

      Addition and multiplication are actually more complex:

      (Once again from my notes)

      // Addition
      const Suc = n => f => x => f(n(f)(x));
      const Add = m => n => m(Suc)(n);
      // Yep, multiplication is just function composition!
      // mnx.m(n(x)) - apply n, m times to x.
      const Mul = Compose;
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        Exponents are just numbers applied to each other.

        How is “apply” defined here?

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          Numbers are just functions. 0 = (f x.x), 1 = (f x.f x), 2 = (f x.f f x) and so on.

          So by apply I mean: (f x.f (f x))(f x.f x) or “2 apply 1” or “1 applied to 2” or “2(1)” :)

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            Here is 2^2. Doing it by hand is really, really difficult because of renaming and what not. You can see it gets confusing very quickly!

            (x.(fy.f(fy))(m.x(x m)))
            (x.(y.(m.x(x m))(x(x y))))
            (x.(y.(x(x (x(x y))))))

            But we see the end result is indeed “4”.

            You can see the gist of the process though: both sides “grow” and then the right side is embedded into the left side.

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              In more of a lambda-calculus sense.

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            But if you look at the Taylor series of the exponential function centered at one of those points, say x = 20, then we have

            exp(-x) = e^{-20} - e^{-20} (x - 20) + e^{-20}/2 (x - 20)^2 + ...

            and all those terms are very, very small close to x = 20.

            To me, this highlights that Taylor series expansions are primarily local tools, and that if you look at what happens to them far away from the point they’re centered around, you’re going to have a bad time.

            If there’s a miracle here, I suppose it is that the Taylor series of entire functions converge everywhere, which happens because of the Cauchy theorem and uniform continuity. Those things are 100% miracles, to be sure.

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              Isn’t the example that the author gives only true when x ~ 0? A simple counter example would be to estimate the polynomial when x = 100 (is 100 a reasonable large number? too large?). You can see that the polynomial expansion gives 4004901, which is definitely not close to 0.

              The real “miracle” in the post is the the Taylor polynomial expansion, which allows you to represent any function (not just an exponential function) by only knowing its value (and the value of some of its derivatives) at any point. The more derivatives you know, the large the interval where the approximation will “work”.

              If we want to talk about the “miracles” of the exponential function, why not bring up Euler’s identity?

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                What the author is talking about happens for every real entire function near a point where it is (a) close to zero and (b) the Taylor expansion is not centered at that point.

                Whether that makes the exponential function less special or Taylor expansions more magical depends on whether you’re a glass half empty or full kind of person.