Taylor Series

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One analogy:

In my head, I see each term in a Taylor series as a slight adjustment to a path, little correction factors to get you closed to the true event. For example, with Leap Years, we memorize that they happen every 4 years. Well, that’s close, but a little too often, so we need to ignore Leap Years that happen on the century mark. But that’s actually a bit too low, so we add them in for years happening every 400 years (2000, 1600, 1200, etc.). And… that’s probably too often, so maybe every 2000 years we take it out. And that’s too low, so every 40,000 years we add them back in…

Another analogy is like driving a car on an icy road, you might turn left, but that overswings, so you turn right, but that overswings, and so on. Many functions can be handled with a given number of corrections (a polynomial) but really “wiggly” paths never stop changing and need an infinite number of corrections to stay on track. You might have a few correction terms which takes you pretty far, but eventually you’ll get off course.

Let’s model a function,

  1. Taking into account constant effects: $f(x)$

  2. Taking into account first-order effects (speed). The function changes as $f’(x)$, so the the total impact of all first-order effects is $\int f’(x) dx = f’(x)x$

  3. Taking int account all second-order effects (changes in speed). These effects are modeled by $f’’(x)$ so we have $\int \int f’’(x) dx = \frac{1}{2} x^2 f’’(x)$

I like writing the $\frac{1}{2}x^2$ separate from $f’’(x)$ so you can see where the integration constants get applied. Basically, we are modeling the function with greater accuracy because we are accounting for “first-order effects” (changes) and second-order effects (changes to the changes) and third order effects (changes to the changes to the changes) and so on.

Make a human-readable sentence out of it. Use balance, income, raise, etc. as an analogy.

f(x) = starting point + all effects from “income” + all effects from raises + all effects from the raise on the raise + …

Eventually we have to see how each effect ties back to the starting value.