Mathematics
Power Series and Taylor Expansion of Functions
Quick fact
Taylor series can compute functions like sine, cosine, and exponential to high accuracy using only arithmetic operations—addition, multiplication, and division—which is exactly what calculators and computers do.
Why this is interesting
Ever wondered how your calculator finds the sine of 30° without consulting a table? The answer lies in an astonishing trick: any smooth function can be rewritten as an infinite polynomial.
Read the full explanation
Understanding Power Series and Taylor Expansion of Functions
Imagine you're trying to draw a curve using only straight-line segments and you want to match it perfectly at one point. You start with the height at that point, then you add a slope to match the tilt, then you add a twist to match the curvature, and so on. A Taylor series does exactly this mathematically: it uses a function's derivatives (which encode slope, curvature, and higher-order 'twists') at a single point to construct a polynomial that becomes a better and better approximation of the function as you add more terms. This is like an infinite polynomial that can mimic the function's behavior around that point.
A deeper explanation
The Taylor series of a smooth function f centered at a is given by f(x) = f(a) + f'(a)(x-a) + f''(a)/2! (x-a)^2 + f'''(a)/3! (x-a)^3 + ... Each term uses a higher-order derivative at a and a power (x-a)^n, with the factorial n! ensuring that the derivative of the polynomial at a matches the derivative of f at a. This construction works because if a power series converges, its derivatives at the center are exactly the coefficients of the series. The surprising depth is that this local information—an infinite set of derivatives at one point—can sometimes determine the function everywhere, as long as the series converges. This is why functions like e^x and sin(x) can be expressed as series that converge for all real x. The series also underpins numerical methods: truncating the series gives a polynomial whose error can be bounded using Taylor's theorem. This concept is fundamental in analysis, both real and complex, where analytic functions are exactly those that locally equal their Taylor series.