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发表于 2021-7-14 07:44
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题:试证\(\,(0< a_{k+1}< a_k < \sum a_n < \infty\;(\forall k))\implies( na_n\to 0).\)
证:据Cauchy准则,对\(\,\varepsilon\small>0,\,\)有\(\,m_{\varepsilon}\small\in\mathbb{N}\,\)使\(\,a_{\lfloor\frac{n}{2}\rfloor}{\small+\cdots+}a_n< \varepsilon\,\small(n>2m_{\varepsilon})\)
\(\qquad\)因\(\{a_n\}\)递减,\(\;\frac{1}{2}na_n< \varepsilon\;(\forall n>2m_{\varepsilon}).\;\;\therefore\;\;\displaystyle{\lim_{n\to\infty}na_n =\small 0.}\small\;\;\square\) |
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