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定理\(\,\star\,\)\(\quad{\Large\frac{c_n}{b_n}}\to A\implies {\Large\frac{c_1,+\cdots+c_n}{b_1+\cdots+b_n}}\to A.\small\;\;(b_k>0,\,b_1+\cdots+b_n\to\infty)\)
证明 因为\(\displaystyle\lim_{n\to\infty}\small\frac{c_n}{b_n}=A,\;\)对\(\small\,\alpha< A< \beta\),有\(\,m\,\)使\(\small\,n>m\,\)时\(\alpha b_n{\small< }c_n{\small<}\beta b_n\)
\(\qquad\)于是\(\;\alpha< {\Large\frac{c_m+\cdots+c_n}{b_m+\cdots+b_n}}< \beta\;\;(n>m).\) 令\(\,n\to\infty\),由\(\,\alpha,\beta\)
\(\qquad\)可任意靠近\(A\) 知道\({\Large\frac{c_m+\cdots+c_n}{b_m+\cdots+b_n}}\to A\), 进而得
\(\underset{\,}{\qquad}{\Large\frac{c_1+\cdots+c_n}{b_1+\cdots+b_n}}={\Large\frac{\frac{c_1+\cdots+c_{m-1}}{b_m+\cdots+b_n}+\frac{c_m+\cdots+c_n}{b_m+\cdots+b_n}}{\frac{b_1+\cdots+b_{m-1}}{b_m+\cdots+b_n}+1}}\to {\large\frac{0+A}{0+1}}=A.\quad\small\square\)
例：对序列\(\{a_n\}\;(a_1=1,a_{n+1}=\ln(1+a_n)),\,\)令,\(\tau(n)=n-\large\frac{2}{a_n}\underset{\,}{,}\)
\(\quad\)据Taylor定理得\(\;\displaystyle\lim_{n\to\infty}{\small\frac{\tau(n+1)-\tau(n)}{\ln(n+1)-\ln n}}=\lim_{n\to\infty}{\small\frac{a_n/6+O(a_n^2)}{\ln(1+\frac{1}{n})}}\,\overset{na_n\to 2}{=\hspace{-3px}=}\,\small\frac{1}{3},\)
\(\quad\)故\(\;\displaystyle\underset{\,}{\lim_{n\to\infty}}{\small\frac{\tau(n)}{\ln(n)}}=\lim_{n\to\infty}{\small\frac{\tau(n)-\tau(1)}{\ln(n)}}\,\overset{\star}{=}\,\lim_{n\to\infty}{\small\frac{\sum_{k=1}^{n-1}(\tau(k+1)-\tau(k))}{\sum_{k=1}^{n-1}(\ln(k+1)-\ln k)}}=\small\frac{1}{3},\)
\(\quad\)由此立即得\(\;\displaystyle\lim_{n\to\infty}{\small\frac{n(na_n-2)}{\ln n}=}\lim_{n\to\infty}{\small\frac{na_n\tau(n)}{\ln n}=\frac{2}{3}}.\quad\small\square\)
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发表于 2020-11-13 11:09