The Real Exponential Function

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

with factorials and the convention $u^{0}=1$. The series converges for every $u\in$ $\mathbb{R}$: 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 Cauchy sequence and converge by Every Cauchy Sequence of Real Numbers Converges; the same comparison shows the series of absolute values converges, so the convergence is absolute. One writes $e=\exp(1)$.