## Sekai 🌐 🗺

Sekai is the kanji for 世界, meaning “the world”. That’s a great word because of the scale it designates.

### TikZ figures for some graphical proofs of this inequality

Goal Show that $$\ln (x) \le x - 1$$ for all $x > 0$. Proof by inverse functions Think about their reflection along the line $y = x$ (i.e. their inverse function). Then we get $$e^x \ge x + 1$$ for all $x \in \Bbb{R}$, which is obviously true. Proof by definite integrals Here’s a second proof using definite integrals. Case 1: x > 1 The rectangle $[1,x] \times [0,1]$ contains the region under the graph of the reciprocal $y = 1/x$ in the domain $[1,x]$. [Read More]

## A Plot for Euler's Constant

### My first PGF Plot for step functions

I firsted tried with foreach, but that would create so much paths. I found them difficult to operate on later, for example, with PGF plots library fillbetween. const plot is a better solution. \documentclass{standalone} \usepackage{pgfplots} \pgfplotsset{compat=1.18} \usetikzlibrary{patterns} \usepgfplotslibrary{fillbetween} \begin{document} \newcommand\myN{8} \pgfplotsset{ axis lines=center, legend style={at={(1,1)},anchor=north east,fill=none}, title style={at={(0.5,1.05)}}, every axis x label/.style={ at={(ticklabel* cs:1)}, anchor=west, }, every axis y label/.style={ at={(ticklabel* cs:1)}, anchor=south, }, } \begin{tikzpicture} \begin{axis}[ title={sum of hatched region converges to Euler's constant}, xlabel={$x$}, ylabel={$y$}, xmin=0, xmax={\the\numexpr\myN+2}, ymin=0, ymax=1. [Read More]