2016-2017第一学期数学分析3-3期中考试

一、计算下列积分

(1)\(\displaystyle \int_{L}\sqrt{x^2+y^2}\mathrm{d}s\)，\(L:x^2+y^2=ax\).

(2)\(\displaystyle \int_{0}^{+\infty}e^{-ax}\cos bx \mathrm{d}x\quad (a>0)\).

二、判断下列级数的收敛性

(1)\(\displaystyle \sum_{n=1}^{\infty}\frac{3^nn!}{n^n}\).

(2)\(\displaystyle \sum_{n=1}^{\infty}\bigg(\frac{1}{n}-\ln (1+\frac{1}{n})\bigg)\).

三、\(S\)是锥面\(z=\sqrt{x^2+y^2}\)在圆柱\(x^2+y^2\leqslant 2x\)内的部分，求\(\displaystyle \iint_{S}z\mathrm{d}S\).

四、判断级数\(\displaystyle \sum_{n=1}^{\infty}\frac{\cos n}{\sqrt[n]{2}(n+1)}\)的敛散性.

五、设\(D\)是\(\mathbb{R}^2\)上带有逐段光滑边界的有界闭区域，\(\Delta =\dfrac{\partial^2}{\partial x^2}+\dfrac{\partial^2}{\partial y^2}\)为拉普拉斯算子.若\(u\in C^2(D),v \in C^1(D)\)，\(\dfrac{\partial u}{\partial \vec{n}}\)表示\(u\)沿\(\partial D\)关于区域\(D\)的外法向量导数，求证

\[\iint_{D}v\Delta u\mathrm{d}x\mathrm{d}y=-\iint_{D}\bigg(\frac{\partial v}{\partial x}\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}\frac{\partial u}{\partial y}\bigg)\mathrm{d}x\mathrm{d}y+\int_{\partial D}v \frac{\partial u}{\partial \vec{n}}\mathrm{d}s.\]

六、\(\vec{F}(x,y,z)=f(r)\vec{r}\)，其中\(r=\sqrt{x^2+y^2+z^2}(r\neq 0)\)，\(\vec{r}=(x,y,z)\)，证明\(\displaystyle \int_{L}\vec{F}\cdot\mathrm{d}\vec{s}\)与路径无关.

七、\(\displaystyle \int_{a}^{+\infty}f(x)\mathrm{d}x\)收敛，\(\dfrac{f(x)}{x}\)在\([a,+\infty)\)上单调递减趋于0.证明\(\lim\limits_{x\rightarrow +\infty}xf(x)=0\).

八、记\(p_n\)表示第\(n\)个素数.证明\(\displaystyle \prod_{n=1}^{\infty}\frac{1}{1-\frac{1}{p_n}}=+\infty\).