1. 求级数 $$ex vsm{n}frac{(-1)^{n-1}}{(2n-1)(2n+1)} eex$$ 的和.
解答: 考虑级数 $$eex ea vsm{n}(-1)^n frac{x^{2n-1}}{(2n-1)(2n+1)} &=vsm{n}frac{(-1)^n}{2n+1}int_0^x t^{2n-2} d t\ &=int_0^x vsm{n}frac{(-1)^n}{2n+1}t^{2n-2} d t\ &=int_0^x frac{1}{t^3} sex{vsm{n} (-1)^n int_0^t s^{2n} d s} d t\ &=int_0^x frac{1}{t^3} sex{int_0^t vsm{n} (-s^2)^n d s} d t\ &=int_0^x frac{1}{t^3}sex{int_0^t frac{-s^2}{1+s^2} d s} d t\ &=int_0^x frac{arctan t-t}{t^3} d t\ &=-frac{1}{2}int_0^x (arctan t-t) d frac{1}{t^2}\ &=-frac{1}{2}sez{frac{arctan x-x}{x^2}-int_0^xfrac{-1}{1+t^2} d t}\ &=frac{x-arctan x-x^2arctan x}{2x^2}\ &equiv f(x),quad xin [-1,1]. eea eeex$$ 于是原级数 $$ex =f(-1)=frac{pi}{4}-frac{1}{2}. eex$$
2. 已知 $S:x^2+y^2+z^2=R^2$, $h eq R$. 试求 $$ex iint_S frac{ d S}{sqrt{x^2+y^2+(z-h)^2}}. eex$$
解答: $$eex ea mbox{原积分}&=iint_Sfrac{ d S}{sqrt{R^2-2hz+h^2}}\ &=iint_{x^2+y^2leq R^2} sex{frac{1}{sqrt{R^2+h^2-2hsqrt{R^2-x^2-y^2}}} +frac{1}{sqrt{R^2+h^2+2hsqrt{R^2-x^2-y^2}}}}\ &quadcdot frac{R}{sqrt{R^2-x^2-y^2}} d x d y\ &=2pi Rint_0^R sex{ frac{1}{sqrt{R^2+h^2-2hsqrt{R^2-r^2}}} +frac{1}{sqrt{R^2+h^2+2hsqrt{R^2-r^2}}}}frac{1}{sqrt{R^2-r^2}}r d r\ &=2pi Rint_0^R sex{frac{1}{sqrt{R^2+h^2-2hs}}+frac{1}{sqrt{R^2+h^2+2hs}}} d squadsex{sqrt{R^2-r^2}=s}\ &equiv 2pi R[f(h)+f(-h)]. eea eeex$$ 为此, 先计算 $$eex ea f(h)&=int_0^R frac{1}{sqrt{R^2+h^2-2hs}} d s\ &=int_{sqrt{R^2+h^2}}^{|R-h|}frac{1}{t}cdot frac{-t}{h} d tquadsex{sqrt{R^2+h^2-2hs}=t}\ &=frac{1}{h}sex{sqrt{R^2+h^2}-|R-h|}. eea eeex$$ 故 $$ex mbox{原积分} =2pi Rsex{frac{sqrt{R^2+h^2}-|R-h|}{h} +frac{sqrt{R^2+h^2}-|R+h|}{-h}} =frac{2pi R}{h}sex{|R+h|-|R-h|}. eex$$
3. 已知 $f$ 非线性, 试证: $$ex sup_{xinbR}|f'(x)|^2leq 2sup_{xinbR} |f(x)|cdot sup_{xinbR}|f''(x)|. eex$$
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 $$ex sev{a{cccc} 1&1&cdots&1\ lm_1&lm_2&cdots&lm_r\ vdots&vdots&&vdots\ lm_1^{r-2}&lm_2^{r-2}&&lm_r^{r-2}\ lm_1^r&lm_2^r&cdots&lm_r^r ea} eex$$ 的值.
(3). 设 $A$ 是 $n$ 阶方阵, 试证: $A$ 幂零 $lra$ $ r(A^p)=0, 1leq pleq n$.
(4). 定义 $[A,B]=AB-BA$, 试证: $$ex [[A,B],A]=0,quad [[A,B],B]=0 a [A,B]mbox{ 幂零}. eex$$
(5). 证明: $$ex [[A,B],C]+[[B,C],A]+[[C,A],B]=0. eex$$
证明: 多写几下即可.