证明数列n+1/n^2+1是无穷小量
\u6570\u5217\u6781\u9650\u8bc1\u660e\uff1a (2\u7684n\u6b21\u65b9-n\uff09\u5206\u4e4b\u4e00\u662f\u65e0\u7a77\u5c0f\u91cf\u8bc1\u660e\uff1a\u30101\u3011\u6613\u77e5\uff0c\u5f53n\u22653\u65f6\uff0c\u6052\u6709\uff1a
n\uff1c(2^n)-n\uff1c2^n.
\u22341/(2^n) \uff1c1/[(2^n)-n] \uff1c1/n..(n=3,4,5,6,\u2026\u2026.).
\u30102\u3011\u6613\u77e5\uff0c\u5f53n----+\u221e\u65f6\uff0c2^n--+\u221e
\u2234 \u5f53n--+\u221e\u65f6\uff0c\u5c31\u67090\uff1c1/2^n\uff1c1/[(2^n)-n] \uff1c1/n.
\u2234\u7531\u201c\u5939\u903c\u5b9a\u7406\u201d\u53ef\u77e5\uff0cIim1/[(2^n)-n]=0.(n-+\u221e).
\u53731/[(2^n)-n]\u662f\u201c\u5f53n-+\u221e\u65f6\u201d\u7684\u65e0\u7a77\u5c0f\u3002
\u8bc1\uff1a\u56e0\u4e3alim|a(n+1)/a(n)|=c\u30081\uff0c\u5373\u6570\u5217|a(n+1)/a(n)|\u6536\u655b
\u7531\u6536\u655b\u6570\u5217\u7684\u5c40\u90e8\u6709\u754c\u6027\uff0c\u30e7N\u2208N+\uff0c\u5f53n>N\u65f6\uff0c
|a(n)/a(n-1)|\u2264(1+c)/2<1\uff0c\u53730\u2264|a(n)|\u2264(1+c)/2*|a(n-1)|
\u7531\u6b640\u2264|a(n)|\u2264[(1+c)/2]^(n-N)*|a(N)|\uff0c\u663e\u7136lim[(1+c)/2]^(n-N)*|a(N)|=0
\u7531\u5939\u903c\u5b9a\u7406\uff0clim|a(n)|=0\uff0c\u5373lima(n)=0
\u6240\u4ee5\u6570\u5217a(n)\u662f\u65e0\u7a77\u5c0f\u6570\u5217
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