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注记:楼上定义的$\,M(a,b)\,$叫作$\,a,\,b\,$的算术几何平均.1: $\qquad\;\;$(Arithmetic-Geometric Mean, AGM)P ©Elinkage数学论坛 -- Elinkage极酷超级论坛 -/5Lbp $\because\;x^2-y^2=(x-y)^2+2y(x-y)$D} 对$\,a> b>0.\;$我们有$\;0< a_{n+1}-b_{n+1}=\small\displaystyle\frac{(\sqrt{a_n}-\sqrt{b_n})^2}{2}< \frac{a_n-b_n}{2}$&HEx $\therefore\quad\boxed{\small\; 0< a_n-b_n < \frac{a-b}{2^n}\,(n>0)\;}\quad(\{a_n\}\,$收敛到$\,M(a,b)\,$极快$).$8N.n( ©Elinkage数学论坛 -- Elinkage极酷超级论坛 \F
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