武汉大学2015年基础数学复试笔试试题 - Eufisky - The lost book
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武汉大学2015年基础数学复试笔试试题

Eufisky posted @ 2016年3月24日 15:51 in 考研 with tags 考研 武汉大学 , 905 阅读

武汉大学2015年基础数学复试笔试试题

1.导函数极限定理, $f'(0)$存在, 而$\lim_{x\to0} f'(x)$不存在的例子.

事实上,可以考察

\[f\left( x \right) = \begin{cases}{x^2}\sin \frac{1}{x}, &x \ne 0\\0, &x = 0\end{cases}.\]

 

2.$\{a_n\}$是正项数列且单增.证明: $\sum_{n = 1}^\infty {\left( {\frac{{{a_{n + 1}}}}{{{a_n}}} - 1} \right)}$收敛$\Leftrightarrow$ $\{a_n\}$有界.

 

3.设$A$是$n$阶可逆复方阵，证明存在分解

\[A=UT,\]

其中$U$是酉矩阵，$T$是主对角线上都是正数的上三角型矩阵，并证明这种分解的唯一性。

 

4.讨论微分方程过点y=0的解的存在性和唯一性，其中$\alpha>0$.

\[\frac{dy}{dx}=|y|^{\alpha}.\]

 

5.证明含参变量积分

\[\int_{0}^{+\infty}\frac{\sin{xy}}{y(1+x)}dy\]

关于$x$在$0<\delta\leq x<+\infty$上一致收敛，在$0<x<+\infty$上非一致收敛。

 

 

6.利用数学归纳法证明$n$维空间中的$n+1$面体${B_{n + 1}}:\sum\limits_{i = 1}^n {{{\left( {{x_i}} \right)}^{1/\alpha }}} \le 1,\alpha > 0$的体积为\[V = \frac{{{2^n}{\alpha ^{n - 1}}{{\left[ {\Gamma \left( \alpha \right)} \right]}^{n - 1}}\Gamma \left( {\alpha + 1} \right)}}{{\Gamma \left( {\alpha n + 1} \right)}},\]

其中$\Gamma$为伽马函数.

 

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