AS证明题～n+1和n是角标ifasequenceofvaluesgivenbytheiterativeformulaX(n+1)=(ln9-lnXn)/2converges,thenitconvergestotherootoftheequationx=9e^(-2x)

Question

AS 证明题～
n+1 和 n是角标
if a sequence of values given by the iterative formula X(n+1)=(ln9-lnXn)/2 converges,then it converges to the root of the equation x=9e^(-2x)

Answer 1

因为Xn的极限存在,设其为X,又ln是连续函数,故
对X(n+1)=(ln9-lnXn)/2两边同时取极限得到
X=(ln9-lnX)/2,整理得
2X=ln(9/X),
e^(2X)=9/X,即 X=9e^(-2X).