June 10, 2025
Fasten your seatbelt.
By Taylor’s theorem, every sufficiently well-behaved function on the real numbers has a power series representation f(x) = \sum_{n=0}^\infty f^{n}(0) x^n/n! around x = 0, where f^{n}(x) denotes the n-th derivative of f. Perhaps the most well-known is the exponential function with power series representation \exp(x) = \sum_{n=0}^\infty x^n/n!.