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I've never forgotten how to make a Taylor series, but I did forget why or where it came from. I thought about it a bit in terms of increasingly accurate approximations with derivatives, then gave up and looked it up.
"Assume a function has a power series in (x-x0), then plug in x=x0 and take derivatives to find the coefficients."
Hmmph. It certainly works, but feels kind of out of thin air.
As for what they're for, my first thought was "deriving wacky identities for pi and e". More seriously, calculating transcendental functions. But I'd not learned one twist on the latter: if you have a nasty integral, you can expand the integrand as a Taylor series, integrate *that*, and voila, you can calculate the integral.
***
As hinted at before, I like revisiting fundamentals. Feynman's lectures chapter 1-22 is like the bible of this, where he starts from natural numbers, goes through making log tables, then evaluating complex exponentials, and finally getting to Euler's formula through sheer calculation. But there are basics he doesn't cover, like evaluating trig functions without Taylor series. The obvious thing is to work from known angles with the half-angle and angle-addition formulas. But where do the raw values come from? How do we know what angles go with what side length ratios?
Well, the isosceles right triangle is entirely determined by its name, that one's trivial. But how do we know that 30-60-90 goes with 1-sqrt(3)-2? I came up with one way: cos(30)=sin(60)=2*sin(30)*cos(30) -> sin(30) = 1/2.
I learned just the other day that sin(18) has a fairly simple algebraic value. But if you try a similar approach to the above, cos(18)=sin(72), you end up with a cubic equation in sin(18). Bleah! Instead there's a geometrical approach, which I can't describe well without diagrams, but it starts with assuming there's an isosceles triangle such that bisecting one of the equal angles yields a similar sub-triangle. That quickly gives you a 36-72-72 triangle; more cleverness yields that the side/base is phi (golden ratio), and so sin(18) = 1/(2phi).
Speaking of which, the proof that phi is irrational is simple and neat. The golden ratio is defined by n/m = m/(n-m), n>m. Assume it's a rational, so that m and n are integers, and n/m has been reduced to lowest terms. But the golden ratio definition requires there be a yet smaller m/(n-m), contradicting the lowest terms assumption... There's a related geometrical proof, where you start with a golden rectangle with integer sides, cut out a square with integer sides, and keep going... you run out of integers.
"Assume a function has a power series in (x-x0), then plug in x=x0 and take derivatives to find the coefficients."
Hmmph. It certainly works, but feels kind of out of thin air.
As for what they're for, my first thought was "deriving wacky identities for pi and e". More seriously, calculating transcendental functions. But I'd not learned one twist on the latter: if you have a nasty integral, you can expand the integrand as a Taylor series, integrate *that*, and voila, you can calculate the integral.
***
As hinted at before, I like revisiting fundamentals. Feynman's lectures chapter 1-22 is like the bible of this, where he starts from natural numbers, goes through making log tables, then evaluating complex exponentials, and finally getting to Euler's formula through sheer calculation. But there are basics he doesn't cover, like evaluating trig functions without Taylor series. The obvious thing is to work from known angles with the half-angle and angle-addition formulas. But where do the raw values come from? How do we know what angles go with what side length ratios?
Well, the isosceles right triangle is entirely determined by its name, that one's trivial. But how do we know that 30-60-90 goes with 1-sqrt(3)-2? I came up with one way: cos(30)=sin(60)=2*sin(30)*cos(30) -> sin(30) = 1/2.
I learned just the other day that sin(18) has a fairly simple algebraic value. But if you try a similar approach to the above, cos(18)=sin(72), you end up with a cubic equation in sin(18). Bleah! Instead there's a geometrical approach, which I can't describe well without diagrams, but it starts with assuming there's an isosceles triangle such that bisecting one of the equal angles yields a similar sub-triangle. That quickly gives you a 36-72-72 triangle; more cleverness yields that the side/base is phi (golden ratio), and so sin(18) = 1/(2phi).
Speaking of which, the proof that phi is irrational is simple and neat. The golden ratio is defined by n/m = m/(n-m), n>m. Assume it's a rational, so that m and n are integers, and n/m has been reduced to lowest terms. But the golden ratio definition requires there be a yet smaller m/(n-m), contradicting the lowest terms assumption... There's a related geometrical proof, where you start with a golden rectangle with integer sides, cut out a square with integer sides, and keep going... you run out of integers.