Anyone have insights how to come up with this inequality? I can prove this but need some insights how people come up with this. It's Chapter 1 question 2(b) in notes http://www-math.mit.edu/~rigollet/PDFs/RigNotes17.pdf
2026-04-08 14:34:24.1775658864
$\frac{e^{-x}}{\sqrt{1-2x}} \leq e^{\frac{x^2}{1-2x}}$ where $0<x<1/2$
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HINT: (Use $2x = t$). Taking logs, on one side is $-1/2 \log(1-t)$, on the other a fraction in $t$. The Taylor exansion at $0$ for LHS is know, for RHS follows easily from decomposing into simple fractions. Now compare the coefficients and notice those on RHS larger for every power of $t$.