AI Tutor
Your calculator evaluates sin⁑(0.42)\sin(0.42) in milliseconds, even though microchips can technically only perform basic arithmetic like addition and multiplication.
The secret is turning complex, curvy functions into infinite polynomialsβ€”expressions made strictly of numbers, variables, and powers.

Power Series Representation

A power series is an infinite polynomial written in ascending powers of xx, with the general form c0+c1x+c2x2+c3x3+…c_0 + c_1x + c_2x^2 + c_3x^3 + \dots, where each cnc_n is a constant.
If we set an unknown function f(x)f(x) equal to this series, our challenge is finding the exact coefficient recipe for every single ara_r.
How do we pinpoint these mystery coefficients using only calculus?

Deriving the Formula

Scottish mathematician Colin Maclaurin published this systematic method in 1742 to standardize calculus derivations pioneered by Isaac Newton and Brook Taylor.
Start by assuming f(x)=a0+a1x+a2x2+a3x3+…f(x) = a_0 + a_1 x + a_2 x^2 + a_3 x^3 + \dots and evaluate at x=0x = 0. Every term with an xx vanishes, leaving f(0)=a0f(0) = a_0.
Next, take the first derivative: fβ€²(x)=a1+2a2x+3a3x2+4a4x3+…f'(x) = a_1 + 2a_2 x + 3a_3 x^2 + 4a_4 x^3 + \dots. Substituting x=0x = 0 gives fβ€²(0)=a1f'(0) = a_1.
Differentiating repeatedly yields higher derivatives where repeated power-rule differentiation gives drdxr(xr)∣x=0=r!\left.\frac{d^r}{dx^r}(x^r)\right|_{x=0} = r!, meaning the coefficients arise naturally as ar=f(r)(0)r!a_r = \frac{f^{(r)}(0)}{r!} .