引用:
原帖由 addcinabo 於 2010-9-21 09:13 AM 發表
第二題的解法超酷的耶,請問有什麼知識背景嗎?
\(\displaystyle f(x)=\left(x-\alpha\right)\left(x-\beta\right)\left(x-\gamma\right)\Rightarrow f\,'(x)=\left(x-\beta\right)\left(x-\gamma\right)+\left(x-\alpha\right)\left(x-\gamma\right)+\left(x-\alpha\right)\left(x-\beta\right)\)
因此,
\(\displaystyle\frac{f\,'(x)}{f(x)}=\frac{1}{x-\alpha}+\frac{1}{x-\beta}+\frac{1}{x-\gamma}\)
\(\displaystyle=\frac{1}{x}\left(1+\frac{\alpha}{x}+\frac{\alpha^2}{x^2}+\cdots\right)+\frac{1}{x}\left(1+\frac{\beta}{x}+\frac{\beta^2}{x^2}+\cdots\right)+\frac{1}{x}\left(1+\frac{\gamma}{x}+\frac{\gamma^2}{x^2}+\cdots\right)\)
\(\displaystyle=3\cdot x^{-1}+\left(\alpha+\beta+\gamma\right)x^{-2}+\left(\alpha^2+\beta^2+\gamma^2\right)x^{-3}+\cdots\)
其中,幾何級數的收斂條件是 \(\displaystyle\left|\frac{\alpha}{x}\right|<1,\left|\frac{\beta}{x}\right|<1,\left|\frac{\gamma}{x}\right|<1\)。