The \textbf{exponential function} $\exp:\mathbb{R}\to\mathbb{R}$ is defined by

with \reftext{def:factorial-natural-number-2026a}{factorials} and the convention $u^{0}=1$. The series converges for every $u\in$ \reftext{def:real-numbers-c54-2026c}{$\mathbb{R}$}: for indices $k>2|u|$ the terms are dominated in absolute value by a geometric sequence with ratio $1/2$, so the partial sums form a \reftext{def:cauchy-sequence-real-c54-2026a}{Cauchy sequence} and converge by \ref{thm:cauchy-sequence-converges-real-c54-2026a}; the same comparison shows the series of absolute values converges, so the convergence is absolute. One writes $e=\exp(1)$.