The real exponential function and the number $e$ by a power series

Depends on

• A real power series about a centre, its interval of convergence, and its radius in $[0,+\infty]$
• The factorial $n!$ and the falling factorial $n^{\underline{k}}$, defined by recursion in $\mathbb{N}$
• Integer powers $a^m$
• The canonical natural $\iota(n) = n \cdot 1_F$ of a field
• Canonical naturals are positive and strictly increasing
• Series, partial sums, convergence and the sum, divergence, and the tail series

Used by

• Dobinski's formula expresses the Bell numbers as Bₙ=e⁻¹∑_ℓ≥0ℓⁿ/ℓ! Corollary
• Every continuous f with f(xy)=f(x)+f(y) is f(x)=c log x for a unique c, including c=0 Corollary
• The elementary numerical bound 2<e<3 Corollary
• The exponential is positive and satisfies exp(-x)=1/exp(x) Corollary
• The number e is the unique x>0 satisfying ∫₁ˣdt/t=1 Corollary
• Real powers for positive bases, with the zero-base positive-exponent convention Definition
• The moment generating function M_X(t)=E[e^tX] on a finite probability space Definition
• The six hyperbolic functions and their natural domains Definition
• The standard flat function Definition
• A parameter ledger for the high-girth, high-chromatic alteration proof Example
• Checking the symmetric local-lemma condition for a sparse uniform hypergraph Example
• The log-free product limit (1-2/n)ⁿ→exp(-2) Example
• 1+x≤exp(x) for every real x, hence (1-p)ᵐ≤exp(-mp) Lemma
• A geometric bound for tails of the exponential series Lemma
• Characteristic function of a normal law Lemma
• Chernoff bound for independent bernoulli trials Lemma
• For a uniform random sign ε, E[e^tε]≤ e^t²/2 Lemma
• The exponential series converges absolutely for every real argument Lemma
• exp(z+w)=exp z exp w, and the complex exponential extends the real exponential Theorem
• For every real x, (1+x/n)ⁿ→exp x Theorem
• If k≥1 and n≥3k² 2ᵏ, an n-vertex tournament with property Sₖ exists Theorem
• log is the unique continuous f:(0,∞)→ℝ with f(xy)=f(x)+f(y) and f(e)=1 Theorem
• Morse stability with explicit parameter dependence Theorem
• Picard iteration from 1 produces the exponential partial sums Theorem
• Regular normalized multiplicative Cauchy equations characterize the exponential Theorem
• Standard Maclaurin expansions Theorem
• The exponential dominates every fixed nonnegative integer power at +∞ Theorem
• The exponential is the unique solution of y'=y with y(0)=1 Theorem
• The exponential tends to +∞ at +∞ and to 0 at -∞ Theorem
• The number e is irrational Theorem
• The power-series, product-limit, IVP, functional-equation, and Picard definitions agree Theorem

Dependency tree · two levels

40 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• MIT OpenCourseWare 18.100B Real Analysis, Spring 2025 full lecture notes (standard reference, not scraped)
• J. Lebl, Basic Analysis, Logarithm and Exponential (standard reference, not scraped)