$(1+x)^{1/x}-x^{1/x}$渐近展开

hbghlyj

当$x>1$时$$(1+x)^{x^{-1}}-x^{x^{-1}}>x^{-2}$$

WolframAlpha
How to calculate generalized Puiseux series?

realnumber

$y=x^{\frac{1}{x}}((1+\frac{1}{x})^\frac{1}{x}-1)>x^\frac{1}{x}(1+\frac{1}{x^2}-1)=x^\frac{1}{x}\frac{1}{x^2}>\frac{1}{x^2}$


提个公因式后，用了下伯努利不等式-对任意实数n≥0,和任意实数x≥-1,有 (1+x)^n≥1+nx