Mathematics
How Taylor Series Approximate Functions Using Polynomials
Quick fact
The Taylor series for e^x is 1 + x + x²/2! + x³/3! + ... and it works for all real numbers, but the Taylor series for ln(1+x) only works when |x| < 1 — exactly the kind of surprising limitation that makes series fascinating.
Why this is interesting
You know sin(x) and e^x, but how does your calculator actually compute them? It turns out every smooth function hides an infinite polynomial inside it — and you can unlock it using derivatives.