数列求极限题
\u6c42\u5173\u4e8e\u6570\u5217\u7684\u6781\u9650\u7684\u9898\u8fd9\u662f\u6570\u5206\u7684\u9898\u5427\uff0c\u6709\u7b54\u6848\u7684
\u89e3\u6790\uff1a
\u95ee\u98981\uff1a
\u56e0\u4e3af(x)=2+1/x
\u6240\u4ee5f`(x)=-1/x²<0
\u6240\u4ee5f(x)\u662f\u5355\u8c03\u4e0b\u964d
\u5047\u8bbexnf(x(n-1))
\u7531x(n+1)=f(xn)\u5f97
x(n+1)-xn=f(xn)-f(x(n-1))>0(\u7531\u5047\u8bbe\u5047\u8bbexnf(x(n-1))\uff09
\u5373x(n+1)>xn
\u4e8e\u662f\u6709xn<x(n+1) xn<x(n-1)\uff08\u8fd9\u662f\u5047\u8bbe\u6765\u7684xn<x(n-1))
\u4e8e\u662f\u6709xn\u6bd4x(n+1)\u5c0f\uff0cxn\u6bd4x(n-1)\u5c0f
\u8fd9\u8bf4\u660e{xn}\u4e0d\u5355\u8c03\u4e86\uff0c\uff08\u82e5\u8981\u5355\u8c03\uff0c\u5219xn\u8981\u5728x(n-1)\u4e0ex(n+1)\u4e4b\u95f4\uff09
\u95ee\u98982\u3001
\u8fd9\u91cc\u7528\u4e86\u6570\u5b66\u5f52\u7eb3\u6cd5
\u5047\u8bbelimxn\u5b58\u5728\u4e14\u7b49\u4e8ea\uff0c\u5219\u5b58\u5728\u03b5>0,\u4f7f|xn-a|<\u03b5/4
\u56e0\u4e3a\u6709|x(n+1)-a|<(1/4)|xn-a|
\u4e8e\u662f\u6709|x(n+1)-a|<(1/4)|xn-a|=\u03b5
\u8fd9\u8bf4\u660elimx(n+1)\u4e5f\u6709\u6781\u9650\uff0c\u5e76\u4e14\u4e5f\u7b49\u4e8ea
∴a^(1\2)<X(n+1)<Xn 由单调有界数列必有极限知极限存在
令Xn=X（n+1）得极限为a^(1\2)
或另解X(n+1)-a^(1\2)=[Xn-a^(1\2)]^2\2Xn
X(n+1)+a^(1\2)=[Xn+a^(1\2)]^2\2Xn
∴[X(n+1)-a^(1\2)]\[X(n+1)+a^(1\2)]=[Xn-a^(1\2)]^2\[Xn+a^(1\2)]^2
得[Xn-a^(1\2)]\[Xn+a^(1\2)]={[X1-a^(1\2)]\[X1+a^(1\2)]}^[2^(n-1)]
X1>a^(1\2)则[X1-a^(1\2)]\[X1+a^(1\2)]<1
所以极限为a^(1\2)
2X(n+1)=Xn+a/Xn
根据a^2+b^2>=2ab
2X(n+1)=Xn+a/Xn〉=2根号a
X(n+1)>=根号a
易证明如果Xn=根号a， X(n+1)=根号a
如果Xn>根号a
X(n+1)<Xn
所以Xn最小值为根号a
当n趋于无穷时，可认为X(n+1)=Xn，因为n+1≈n
即Xn=(Xn+a/Xn)/2
解得Xn=√a
即当n趋于无穷时,limXn=√a
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