1. 求级数 $$\bex \vsm{n}\frac{(-1)^{n-1}}{(2n-1)(2n+1)} \eex$$ 的和.
解答: 考虑级数 $$\beex \bea \vsm{n}(-1)^n \frac{x^{2n-1}}{(2n-1)(2n+1)} &=\vsm{n}\frac{(-1)^n}{2n+1}\int_0^x t^{2n-2}\rd t\\ &=\int_0^x \vsm{n}\frac{(-1)^n}{2n+1}t^{2n-2}\rd t\\ &=\int_0^x \frac{1}{t^3} \sex{\vsm{n} (-1)^n \int_0^t s^{2n}\rd s}\rd t\\ &=\int_0^x \frac{1}{t^3} \sex{\int_0^t \vsm{n} (-s^2)^n\rd s}\rd t\\ &=\int_0^x \frac{1}{t^3}\sex{\int_0^t \frac{-s^2}{1+s^2}\rd s}\rd t\\ &=\int_0^x \frac{\arctan t-t}{t^3}\rd t\\ &=-\frac{1}{2}\int_0^x (\arctan t-t)\rd \frac{1}{t^2}\\ &=-\frac{1}{2}\sez{\frac{\arctan x-x}{x^2}-\int_0^x\frac{-1}{1+t^2}\rd t}\\ &=\frac{x-\arctan x-x^2\arctan x}{2x^2}\\ &\equiv f(x),\quad x\in [-1,1]. \eea \eeex$$ 于是原级数 $$\bex =f(-1)=\frac{\pi}{4}-\frac{1}{2}. \eex$$
2. 已知 $S:x^2+y^2+z^2=R^2$, $h\neq R$. 试求 $$\bex \iint_S \frac{\rd S}{\sqrt{x^2+y^2+(z-h)^2}}. \eex$$
解答: $$\beex \bea \mbox{原积分}&=\iint_S\frac{\rd S}{\sqrt{R^2-2hz+h^2}}\\ &=\iint_{x^2+y^2\leq R^2} \sex{\frac{1}{\sqrt{R^2+h^2-2h\sqrt{R^2-x^2-y^2}}} +\frac{1}{\sqrt{R^2+h^2+2h\sqrt{R^2-x^2-y^2}}}}\\ &\quad\cdot \frac{R}{\sqrt{R^2-x^2-y^2}}\rd x\rd y\\ &=2\pi R\int_0^R \sex{ \frac{1}{\sqrt{R^2+h^2-2h\sqrt{R^2-r^2}}} +\frac{1}{\sqrt{R^2+h^2+2h\sqrt{R^2-r^2}}}}\frac{1}{\sqrt{R^2-r^2}}r\rd r\\ &=2\pi R\int_0^R \sex{\frac{1}{\sqrt{R^2+h^2-2hs}}+\frac{1}{\sqrt{R^2+h^2+2hs}}}\rd s\quad\sex{\sqrt{R^2-r^2}=s}\\ &\equiv 2\pi R[f(h)+f(-h)]. \eea \eeex$$ 为此, 先计算 $$\beex \bea f(h)&=\int_0^R \frac{1}{\sqrt{R^2+h^2-2hs}}\rd s\\ &=\int_{\sqrt{R^2+h^2}}^{|R-h|}\frac{1}{t}\cdot \frac{-t}{h}\rd t\quad\sex{\sqrt{R^2+h^2-2hs}=t}\\ &=\frac{1}{h}\sex{\sqrt{R^2+h^2}-|R-h|}. \eea \eeex$$ 故 $$\bex \mbox{原积分} =2\pi R\sex{\frac{\sqrt{R^2+h^2}-|R-h|}{h} +\frac{\sqrt{R^2+h^2}-|R+h|}{-h}} =\frac{2\pi R}{h}\sex{|R+h|-|R-h|}. \eex$$
3. 已知 $f$ 非线性, 试证: $$\bex \sup_{x\in\bbR}|f'(x)|^2\leq 2\sup_{x\in\bbR} |f(x)|\cdot \sup_{x\in\bbR}|f''(x)|. \eex$$
证明: 常见的题目了.
4.
(1). 设 $A$ 是 $n$ 阶方阵. 试证: $A$ 幂零 $\lra$ $A$ 的特征多项式 $f(\lm)=\lm^n$.
(2). 求行列式 $$\bex \sev{\ba{cccc} 1&1&\cdots&1\\ \lm_1&\lm_2&\cdots&\lm_r\\ \vdots&\vdots&&\vdots\\ \lm_1^{r-2}&\lm_2^{r-2}&&\lm_r^{r-2}\\ \lm_1^r&\lm_2^r&\cdots&\lm_r^r \ea} \eex$$ 的值.
(3). 设 $A$ 是 $n$ 阶方阵, 试证: $A$ 幂零 $\lra$ $\tr(A^p)=0,\ 1\leq p\leq n$.
(4). 定义 $[A,B]=AB-BA$, 试证: $$\bex [[A,B],A]=0,\quad [[A,B],B]=0\ra [A,B]\mbox{ 幂零}. \eex$$
(5). 证明: $$\bex [[A,B],C]+[[B,C],A]+[[C,A],B]=0. \eex$$
证明: 多写几下即可.
