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Last edited by Czhang271828 2023-7-6 19:42直接算. 不妨设存在 $m\in \mathbb N_+$ 使得 $\dfrac \beta m<1$. 计算得
\begin{align*}
&\left(1-\dfrac{\beta }{m+1}\right)\left(1-\dfrac{\beta }{m+2}\right)\cdots \left(1-\dfrac{\beta }{m+n}\right)\\[8pt]
=\,&\exp\sum _{k=1}^n\ln \left(1-\dfrac{\beta}{{m+k}}\right)\leq \exp \sum_{k=1}^n\dfrac{-\beta }{m+k}\\[8pt]
\leq \,&\exp \left(-\beta \cdot \int_{m+1}^{m+n+1}\dfrac{\mathrm dx}{x}\right)
\leq \exp \left(\beta\cdot \ln\dfrac{m+1}{m+n+1}\right)\\[8pt]
=\,& \left(\dfrac{m+1}{m+n+1}\right)^\beta.
\end{align*}
从而 $b_n$ 被 $\dfrac{C}{n^\beta }$ 控制. |
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Czhang271828
posted 2023-7-6 13:51
