高等数学（工本）

空间解析几何与向量代数

距离公式

$$\begin{array}{r}|P_1{P}_2|=\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2+(z_2-z_1{)}^2}\end{array}$$

加法

$$\begin{array}{r}\overrightarrow{OA}+\overrightarrow{OB}=\overrightarrow{OC}\\ \overrightarrow{OA}+\overrightarrow{AC}=\overrightarrow{OC}\\ \overrightarrow{OA}-\overrightarrow{OB}=\overrightarrow{BA}\end{array}$$

向量坐标

$$\begin{gathered} \overrightarrow{M_1{M}_2}=\{(x_2-x_1),(y_2-y_1),(z_2-z_1)\} \\ \alpha =\{a1,a2,a3\},\beta =\{b1,b2,b3\} \\ \mathrm{若}\alpha //\beta ，\mathrm{则}\frac{a1}{b1}=\frac{a2}{b2}=\frac{a3}{b3} \end{gathered}$$

数量积

$$\begin{gathered} \alpha \cdot \beta =|\alpha ||\beta |\cos \phi \\ \mathrm{若}\alpha \mathrm{与}\beta \mathrm{垂}\mathrm{直}，\mathrm{则}\alpha \cdot \beta =0 \end{gathered}$$

数量积的坐标表示

$$\begin{array}{r}\alpha =\{a1,a2,a3\},\beta =\{b1,b2,b3\}\\ \alpha \cdot \beta =a_1{b}_1+a_2{b}_2+a_3{b}_3\end{array}$$

平面方程

点法式

$$\begin{array}{r}\text{点}{P}_0(x_0,y_0,z_0),\text{非零向量}n=\{A,B,C\}\\ n\cdot \overrightarrow{P_{0}P}=0\\ A(x-x_0)+B(y-y_0)+C(z-z_0)-0\end{array}$$

一般方程

$$Ax+By+Cz+D=0$$

截距式方程

$$\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1$$

两个平面夹角

$$\begin{array}{r}{n}_1=\{A_1,B_1,C_1\},n_2=\{A_2,B_2,C_2\}\\ \cos \theta =\frac{n_1\cdot n_2}{|n_1||n_2|}=\frac{|A_1{A}_2+B_1{B}_2+C_1{C}_2|}{\sqrt{A_1^2+B_1^2+C_1^2}\sqrt{A_2^2+B_2^2+C_2^2}}\end{array}$$

点到平面距离

$$d=\frac{|A{x}_0+B{y}_0+C{z}_0+D|}{\sqrt{A^2+B^2+C^2}}$$

直线方程

对称式方程

$$\frac{x-x_0}{l}=\frac{y-y_0}{m}=\frac{z-z_0}{n}$$

参数方程

$$\begin{array}{rl}& \frac{x-x_0}{l}=\frac{y-y_0}{m}=\frac{z-z_0}{n}\\ & \{\begin{array}{l}x=x_0+lt,\\ y=y_0+mt,\\ z=z_0+nt,\end{array}\end{array}$$

一般方程

$$\{\begin{array}{l}{A}_{1}x+B_{1}y+C_{1}z+D_1=0,\\ A_{2}x+B_{2}y+C_{2}z+D_2=0,\end{array}$$

直线的夹角

$$\begin{array}{r}{v}_1=\{l_1,m_1,n_1\},v_2=\{l_2,m_2,n_2\}\\ \cos \theta =\frac{v_1\cdot v_2}{|v_1||v_2|}=\frac{|l_1{l}_2+m_1{m}_2+n_1{n}_2|}{\sqrt{l_1^2+m_1^2+n_1^2}\sqrt{l_2^2+m_2^2+n_2^2}}\end{array}$$

直线与平面夹角

$$\begin{array}{r}v=\{l,m,n\},n=\{A,B,C\}\\ \sin \theta =\frac{v\cdot n}{|v||n|}=\frac{|lA+mB+nC|}{\sqrt{l^2+m^2+n^2}\sqrt{A^2+B^2+C^2}}\end{array}$$

多元函数的微分学

偏导数

$$\begin{array}{r}\underset{\mathrm{\Delta }x\to 0}{lim}\frac{\mathrm{\Delta }F}{\mathrm{\Delta }x}=\underset{\mathrm{\Delta }x\to 0}{lim}\frac{f(x_0+\mathrm{\Delta }x,y_0)-f(x_0,y_0)}{\mathrm{\Delta }x}\end{array}$$

全微分

$$dz=f_x(x,y)dx+f_y(x,y)dy$$

复合函数偏导数

$$\frac{dz}{dx}=\frac{\partial z}{\partial u}\cdot \frac{du}{dx}+\frac{\partial z}{\partial v}\cdot \frac{dv}{dx}$$

隐函数偏导数

$$\frac{dy}{dx}=-\frac{F_x}{F_y}$$

偏导数应用

无条件极值

1. 解方程组

$$\{\begin{array}{l}{f}_x(x,y)=0\\ f_y(x,y)=0\end{array}$$

求出$f(x,y)$的全部驻点

2. 对每个驻点计算

$$A=f_{xx}(x_0,y_0),B=f_{xy}(x_0,y_0),C=f_{yy}(x_0,y_0),\mathrm{\Delta }=B^2-AC$$

3. 判断

1. $\mathrm{\Delta }<0$,点$(x_0,y_0)$是函数$f(x,y)$的极值点，当$A<0$时，$f(x_0,y_0)$是极大值，当$A>0$时，$f(x_0,y_0)$是极小值
2. $\mathrm{\Delta }>0$,点$(x_0,y_0)$不是函数$f(x,y)$的极值点
3. $\mathrm{\Delta }=0$,不能判断

条件极值

1. 构造拉格朗日函数

$$L(x,y)=f(x,y)+\lambda \phi (x,y)$$

2. 解方程组

$$\{\begin{array}{ll}\frac{\partial L}{\partial x}& =f_x(x,y)+\lambda {\phi }_x(x,y)=0\\ \frac{\partial L}{\partial y}& =f_y(x,y)+\lambda {\phi }_y(x,y)=0\\ \phi (x,y)& =0\end{array}$$

空间曲线的切线与法平

$$\begin{gathered} \mathrm{空}\mathrm{间}\mathrm{区}\mathrm{线}L：\{\begin{array}{l}x=x(t)\\ y=y(t)\\ z=z(t)\end{array}，L\mathrm{上}\mathrm{的}\mathrm{点}{P}_0(x_0,y_0,z_0) \\ L\mathrm{在}{P}_0\mathrm{的}\mathrm{切}\mathrm{向}\mathrm{量}\mathrm{为}：\{x^{\prime }(t_0),y^{\prime }(t_0),z^{\prime }(t_0)\} \end{gathered}$$

切线方程

$$\frac{x-x_0}{x^{\prime }(t_0)}=\frac{y-y_0}{y^{\prime }(t_0)}=\frac{z-z_0}{z^{\prime }(t_0)}$$

法平面方程

$$x^{\prime }(t_0)(x-x_0)+y^{\prime }(t_0)(y-y_0)+z^{\prime }(t_0)(z-z_0)=0$$

空间曲面的切平面与法线

$$\begin{gathered} \mathrm{曲}\mathrm{面}\mathrm{\Sigma }\mathrm{的}\mathrm{方}\mathrm{程}F(x,y,z)=0，\mathrm{\Sigma }\mathrm{上}\mathrm{的}\mathrm{点}{P}_0(x_0,y_0,z_0) \\ \mathrm{\Sigma }\mathrm{在}{P}_0\mathrm{上}\mathrm{的}\mathrm{法}\mathrm{向}\mathrm{量}：\{F_x(x_0,y_0,z_0),F_y(x_0,y_0,z_0),F_z(x_0,y_0,z_0)\} \end{gathered}$$

法线方程

$$\frac{x-x_0}{F_x(x_0,y_0,z_0)}=\frac{y-y_0}{F_y(x_0,y_0,z_0)}=\frac{z-z_0}{F_y(x_0,y_0,z_0)}$$

切平面方程

$$F_x(x_0,y_0,z_0)(x-x_0)+F_y(x_0,y_0,z_0)(y-y_0)+F_z(x_0,y_0,z_0)(z-z_0)=0$$

方向导数

$$\frac{\partial z}{\partial l}=\frac{\partial z}{\partial x}\cos \alpha +\frac{\partial z}{\partial y}\cos \beta$$

梯度

$$gradf(x_0,y_0)=f_x(x_0,y_0)i+f_y(x_0,y_0)i=\{f_x(x_0,y_0),f_y(x_0,y_0)\}$$

重积分

二重积分性质

性质一

$$\underset{D}{\iint }kf(x,y)d\sigma =k\underset{D}{\iint }f(x,y)d\sigma$$

性质二

$$\begin{gathered} \underset{D}{\iint }(f(x,y)\pm g(x,y))d\sigma =\underset{D}{\iint }f(x,y)d\sigma \pm \underset{D}{\iint }g(x,y)d\sigma \\ \underset{D}{\iint }(af(x,y)\pm bg(x,y))d\sigma =a\underset{D}{\iint }f(x,y)d\sigma \pm b\underset{D}{\iint }g(x,y)d\sigma \end{gathered}$$

性质三

$$\underset{D}{\iint }f(x,y)d\sigma =\underset{D_1}{\iint }f(x,y)d\sigma +\underset{D_2}{\iint }f(x,y)d\sigma$$

性质四

$$\begin{gathered} \mathrm{若}f(x,y)\ge g(x,y) \\ \mathrm{则}\underset{D}{\iint }f(x,y)d\sigma \ge \underset{D}{\iint }g(x,y)d\sigma \end{gathered}$$$$|\underset{D}{\iint }f(x,y)d\sigma |\le \underset{D}{\iint }|f(x,y)|d\sigma$$

性质五

$$\begin{gathered} \mathrm{若}m\le f(x,y)\le M \\ \mathrm{则}m|D|\le \underset{D}{\iint }f(x,y)d\sigma \le M|D|(|D|\mathrm{表}\mathrm{示}\mathrm{区}\mathrm{域}D\mathrm{的}\mathrm{面}\mathrm{积}) \end{gathered}$$

性质六

$$\begin{gathered} \underset{D}{\iint }f(x,y)d\sigma =f(\xi ,\eta )|D| \\ \mathrm{当}f(x,y)=1\mathrm{时}，\mathrm{有}\underset{D}{\iint }f(x,y)d\sigma =\underset{D}{\iint }1d\sigma =\underset{D}{\iint }d\sigma =|D| \end{gathered}$$

直角坐标下的二重积分

$$\begin{array}{l}\begin{array}{lcl}D& & \text{有界闭区域,积分区域}\\ {\mathrm{\Delta }}_{{\sigma }_i}& & \text{第i个小闭区域}\\ \lambda & & \text{所有小闭区域中直径最大的值}\\ f(x,y),g(x,y)& & \text{定义在D上的有界函数,被积函数}\\ \sigma & & \text{D的面积}\\ d\sigma ,dxdy& & \text{直角坐标系中的微元面积}\\ \alpha ,\beta & & \text{常数}\\ m,M& & \text{在D上f(x,y)的最小值和最大值}\end{array}\\ {\iint }_{D}f(x,y)d\sigma =\underset{\lambda \to 0}{lim}\sum _{i=1}^{n}f({\xi }_i,{\eta }_i)\mathrm{\Delta }{\sigma }_i\\ \underset{D}{\iint }f(x,y)dxdy={\int }_a^b[{\int }_{{\phi }_1(x)}^{{\phi }_2(x)}f(x,y)dy]dx\\ \underset{D}{\iint }f(x,y)dxdy={\int }_a^{b}dx{\int }_{{\phi }_1(x)}^{{\phi }_2(x)}f(x,y)dy\end{array}$$

奇偶性

设积分区域 D 关于 y 轴对称，$D=D_{\text{左}}+D_{\text{右}}$

1. 若被积函数 f(x,y)是关于 x 的奇函数，即$f(-x,y)=-f(x,y)$,则$I=\underset{D}{\iint }f(x,y)dxdy=0$
2. 若被积函数 f(x,y)是关于 x 的偶函数，即$f(-x,y)=f(-x,y)$,则$I=\underset{D}{\iint }f(x,y)dxdy=2\underset{D_{\text{左}}}{\iint }f(x,y)dxdy=2\underset{D_{\text{右}}}{\iint }f(x,y)dxdy$

极坐标下的二重积分

$$\underset{D}{\iint }f(x,y)dxdy=\underset{D}{\iint }f(r\cos \theta ,r\sin \theta )rdrd\theta ={\int }_{\alpha }^{\beta }d\theta {\int }_{{\phi }_1(\theta )}^{{\phi }_2(\theta )}f(r\cos \theta ,r\sin \theta )rdr$$

三重积分

直角坐标

$$\begin{array}{l}\begin{array}{lcl}\mathrm{\Omega }& & \text{空间有界闭区域,积分区域}\\ \mathrm{\Delta }{\upsilon }_i& & \text{第i个小闭区域}\\ f(x,y,z)& & \text{定义在}\mathrm{\Omega }\text{上的有界函数,被积函数}\\ \upsilon & & \mathrm{\Omega }\text{的体积}\\ d\upsilon ,dxdydz& & \text{直角坐标系中的微元面积}\end{array}\\ \underset{\mathrm{\Omega }}{\iiint }f(x,y,z)d\upsilon =\underset{\lambda \to 0}{lim}\sum _{i=1}^{n}f({\xi }_i,{\eta }_i,{\zeta }_i)\mathrm{\Delta }\upsilon \\ \underset{\mathrm{\Omega }}{\iiint }f(x,y,z)dxdydz={\int }_a^b({\int }_{{\phi }_1(x)}^{{\phi }_2(x)}({\int }_{z_1(x,y)}^{z_2(x,y)}f(x,y)dz)dy)dx\\ \underset{\mathrm{\Omega }}{\iiint }f(x,y,z)dxdydz=\underset{D_{xy}}{\iint }dxdy{\int }_{z_1(x,y)}^{z_2(x,y)}f(x,y,z)dz\\ \underset{\mathrm{\Omega }}{\iiint }f(x,y,z)dxdydz={\int }_a^{b}dx{\int }_{y_1(x)}^{y_2(x)}dy{\int }_{z_1(x,y)}^{z_2(x,y)}f(x,y,z)dz\end{array}$$

奇偶性

柱面坐标

$$\begin{array}{rl}& \{\begin{array}{l}x=r\cos \theta \\ y=r\sin \theta \\ z=z\end{array}\\ \underset{\mathrm{\Omega }}{\iiint }f(x,y,z)dv& =\underset{\mathrm{\Omega }}{\iiint }f(r\cos \theta ,r\sin \theta ,z)rdrd\theta dz\\ & =\underset{D}{\iint }dxdy{\int }_{z_1(x,y)}^{z_2(x,y)}f(x,y,z)dz\\ & =\underset{D}{\iint }drd\theta {\int }_{\stackrel{\sim }{z_1}(x,y)}^{\stackrel{\sim }{z_2}(x,y)}f(r\cos \theta ,r\sin \theta ,z)rdz\end{array}$$

曲线积分与曲面积分

对弧长的曲线积分

$$\{\begin{array}{l}x=\psi (t),\\ y=\phi (t)\end{array}\alpha \le t\le \beta$$$${\int }_{L}f(x,y)ds={\int }_{\alpha }^{\beta }f(\psi (t),\phi (t))\sqrt{({\psi }^{\prime }(t){)}^2+{\phi }^{\prime }(t)}dt$$

极坐标

$$\begin{gathered} L:r=r(\theta ),{\theta }_0\le \theta \le {\theta }_1 \\ \{\begin{array}{l}x=r(\theta )\cos \theta \\ y=r(\theta )\sin \theta \end{array}({\theta }_0\le \theta \le {\theta }_1) \\ ds=\sqrt{(x^{{}^{\prime }}(\theta ){)}^2+(y^{{}^{\prime }}(\theta ){)}^2}d\theta \\ {\int }_{L}f(x,y)ds={\int }_{{\theta }_0}^{{\theta }_1}f(r(\theta )\cos \theta ,r(\theta )\sin \theta )\sqrt{r^2(\theta )+(r^{{}^{\prime }}(\theta ){)}^2}d\theta \end{gathered}$$

对坐标的曲线积分

$$\{\begin{array}{l}x=\psi (t),\\ y=\phi (t),\\ dx={\psi }^{\prime }(t)dt,\\ dy={\phi }^{\prime }(t)dt\end{array}$$$${\int }_{L_{AB}}=P(x,y)dx+Q(x,y)dy={\int }_{\alpha }^{\beta }(P(\psi (t),\phi (t)){\psi }^{\prime }(t)+Q(\psi (t),\phi (t)){\phi }^{\prime }(t))dt$$

格林公式

$${\iint }_D(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y})dxdy={\oint }_{L}Pdx+Qdy$$

对面积的曲面积分

$$\underset{\mathrm{\Sigma }}{\iint }f(x,y,z)dS=\underset{D_{xy}}{\iint }f(x,y,z(x,y))\sqrt{1+z_x^2+z_y^2}dxdy$$

对坐标的曲面积分

$$\underset{\mathrm{\Sigma }}{\iint }R(x,y,z)dxdy=\{\begin{array}{ll}+\underset{D_{xy}}{\iint }R(x,y,z(x,y))dxdy,& \text{若在}\mathrm{\Sigma }\text{上恒有}0\le \gamma <\frac{\pi }{2}(\text{上侧})\\ -\underset{D_{xy}}{\iint }R(x,y,z(x,y))dxdy,& \text{若在}\mathrm{\Sigma }\text{上恒有}\frac{\pi }{2}<\gamma \le \pi (\text{下侧})\\ 0,& \text{若在}\mathrm{\Sigma }\text{上恒有}\gamma =\frac{\pi }{2}\end{array}$$

常微分方程

一阶微分方程

可分离变量的微分方程

$$\begin{array}{rl}& \frac{dy}{dx}=f(x)g(y)\\ & \frac{dy}{g(y)}=f(x)dx\end{array}$$

齐次方程

$$\begin{gathered} \frac{dy}{dx}=f(x,y) \\ \frac{dy}{dx}=\phi (\frac{y}{x}) \\ \mathrm{令}u=\frac{y}{x},\mathrm{则}y=xu,\frac{dy}{dx}=u+x\frac{du}{dx} \\ u+x\frac{du}{dx}=\phi (u) \\ x\frac{du}{dx}=\phi (u)-u \end{gathered}$$

一阶线性微分方程

$$\begin{array}{rl}& \frac{dy}{dx}+P(x)y=Q(x)\\ & y=(\int Q(x)e^{\int P(x)dx}dx+C)e^{-\int P(x)dx}\end{array}$$

可降阶的二阶微分方程

y''=f(x)

求两次积分

二阶线性微分方程

齐次

$$y^{″}=p(x)y^{\prime }=q(x)y=f(x)$$

非齐次

$$y=\bar{y}(x)+y^{\ast }(x)$$

二阶常系数线性微分方程

$$\begin{gathered} y^{″}+p{y}^{\prime }+qy=0 \\ {r}^2+pr+q=0 \end{gathered}$$$$\begin{array}{rl}& y=C_1{e}^{r_{1}x}+C_2{e}^{r_{2}x}\\ & y=e^{ax}(C_{1}cos\beta x+C_{2}sin\beta x)\\ & y=(C_1+C_{2}x)e^{rx}\end{array}$$

无穷级数

数项级数的概率

$$\begin{gathered} s_n=\sum _{k=1}^n{u}_k=u_1+u_2+u_3+\cdots +u_n \\ \mathrm{若}\mathrm{存}\mathrm{在}\mathrm{常}\mathrm{数}s\mathrm{使}\mathrm{得}:s=\underset{}{lim}{s}_n，\mathrm{则}\mathrm{称}\mathrm{该}\mathrm{级}\mathrm{数}\mathrm{收}\mathrm{敛}，\mathrm{否}\mathrm{则}\mathrm{发}\mathrm{散} \end{gathered}$$

数项级数的基本性质

性质一

$$\begin{gathered} \sum _{n=1}^{\mathrm{\infty }}{u}_n\mathrm{和}\sum _{n=1}^{\mathrm{\infty }}C{u}_n\mathrm{都}\mathrm{收}\mathrm{敛},\mathrm{则} \\ \sum _{n=1}^{\mathrm{\infty }}C{u}_n=C\sum _{n=1}^{\mathrm{\infty }}{u}_n \end{gathered}$$

性质二

$$\begin{gathered} \sum _{n=1}^{\mathrm{\infty }}{u}_n,\sum _{n=1}^{\mathrm{\infty }}{v}_n\mathrm{分}\mathrm{别}\mathrm{收}\mathrm{敛}\mathrm{于}\mathrm{和}s,\sigma ,\mathrm{则} \\ \sum _{n=1}^{\mathrm{\infty }}(u_n\pm v_n)\mathrm{也}\mathrm{收}\mathrm{敛}\mathrm{于}\mathrm{和}s\pm \sigma \end{gathered}$$

性质三

在级数中去掉、增加或改变有限项，其敛散性不变

性质四

$$\begin{gathered} \mathrm{若}\sum _{n=1}^{\mathrm{\infty }}{u}_n\mathrm{收}\mathrm{敛}，\mathrm{则}\mathrm{对}\mathrm{这}\mathrm{个}\mathrm{级}\mathrm{数}\mathrm{的}\mathrm{项}\mathrm{任}\mathrm{意}\mathrm{加}\mathrm{括}\mathrm{号}\mathrm{所}\mathrm{得}\mathrm{的}\mathrm{新}\mathrm{级}\mathrm{数} \\ (u_1+\cdots +u_{n_1})+(u_{n_1+1}+\cdots +u_{n_2})+\cdots +(u_{n_{k-1}+1}+\cdots +u_{n_k})+\cdots \\ \mathrm{也}\mathrm{收}\mathrm{敛}，\mathrm{且}\mathrm{其}\mathrm{和}\mathrm{不}\mathrm{变} \end{gathered}$$

性质五（收敛的必要条件）

$$\mathrm{若}\sum _{n=1}^{\mathrm{\infty }}{u}_n\mathrm{收}\mathrm{敛},\mathrm{则}\mathrm{必}\mathrm{有}\underset{n\to \mathrm{\infty }}{lim}{u}_n=0$$

常见级数

调和级数

$$\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n}\mathrm{发}\mathrm{散}$$

P 级数

$$\begin{gathered} \sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^p}(p>0) \\ p>1\mathrm{时}\mathrm{收}\mathrm{敛}，0<p\le 1\mathrm{时}\mathrm{发}\mathrm{散} \end{gathered}$$

交错级数

$$\sum _{n=1}^{\mathrm{\infty }}(-1{)}^{n-1}{u}_n\mathrm{和}\sum _{n=1}^{\mathrm{\infty }}(-1{)}^n{u}_n\mathrm{为}\mathrm{交}\mathrm{错}\mathrm{级}\mathrm{数}，\mathrm{两}\mathrm{者}\mathrm{敛}\mathrm{散}\mathrm{性}\mathrm{相}\mathrm{同}$$

绝对收敛和条件收敛

$$\begin{gathered} \mathrm{若}\sum _{n=1}^{\mathrm{\infty }}|u_n|\mathrm{收}\mathrm{敛}，\mathrm{则}\mathrm{称}\sum _{n=1}^{\mathrm{\infty }}{u}_n\mathrm{绝}\mathrm{对}\mathrm{收}\mathrm{敛} \\ \mathrm{若}\sum _{n=1}^{\mathrm{\infty }}{u}_n\mathrm{收}\mathrm{敛},\mathrm{而}\sum _{n=1}^{\mathrm{\infty }}|u_n|\mathrm{发}\mathrm{散}，\mathrm{则}\mathrm{称}\sum _{n=1}^{\mathrm{\infty }}{u}_n\mathrm{条}\mathrm{件}\mathrm{收}\mathrm{敛} \end{gathered}$$

审敛法

定理 1

$$\mathrm{正}\mathrm{项}\mathrm{级}\mathrm{数}\sum _{n=1}^{\mathrm{\infty }}{u}_n\mathrm{收}\mathrm{敛}\mathrm{的}\mathrm{充}\mathrm{要}\mathrm{条}\mathrm{件}\mathrm{是}:\mathrm{它}\mathrm{的}\mathrm{部}\mathrm{分}\mathrm{和}\mathrm{数}\mathrm{列}\{s_n\}\mathrm{有}\mathrm{上}\mathrm{界}$$

定理 2（比较审敛法）

$$\begin{gathered} \mathrm{设}\sum _{n=1}^{\mathrm{\infty }}{u}_n\mathrm{和}\sum _{n=1}^{\mathrm{\infty }}{v}_n\mathrm{都}\mathrm{是}\mathrm{正}\mathrm{项}\mathrm{级}\mathrm{数}，\mathrm{且}{u}_n\le v_n(n=1,2,\cdots ) \\ \mathrm{若}\sum _{n=1}^{\mathrm{\infty }}{v}_n\mathrm{收}\mathrm{敛}，\mathrm{则}\sum _{n=1}^{\mathrm{\infty }}{u}_n\mathrm{收}\mathrm{敛}(\mathrm{大}\mathrm{的}\mathrm{收}\mathrm{敛}，\mathrm{小}\mathrm{的}\mathrm{也}\mathrm{收}\mathrm{敛}) \\ \mathrm{若}\sum _{n=1}^{\mathrm{\infty }}{u}_n\mathrm{发}\mathrm{散}，\mathrm{则}\sum _{n=1}^{\mathrm{\infty }}{v}_n\mathrm{发}\mathrm{散}(\mathrm{小}\mathrm{的}\mathrm{发}\mathrm{散}，\mathrm{大}\mathrm{的}\mathrm{也}\mathrm{发}\mathrm{散}) \end{gathered}$$

定理 3（比较审敛法的极限形式）

$$\begin{gathered} \mathrm{设}\underset{n\to \mathrm{\infty }}{lim}\frac{u_n}{v_n}=l\mathrm{和}\sum _{n=1}^{\mathrm{\infty }}{v}_n\mathrm{都}\mathrm{是}\mathrm{正}\mathrm{项}\mathrm{级}\mathrm{数} \\ \mathrm{如}\mathrm{果}\underset{n\to \mathrm{\infty }}{lim}\frac{u_n}{v_n}=l(0<l<+\mathrm{\infty }) \\ \mathrm{则}\sum _{n=1}^{\mathrm{\infty }}{u}_n\mathrm{和}\sum _{n=1}^{\mathrm{\infty }}{v}_n\mathrm{同}\mathrm{收}\mathrm{敛}\mathrm{或}\mathrm{同}\mathrm{发}\mathrm{散} \end{gathered}$$

定理 4【比值审敛法，达朗贝尔（D'Alembert）审敛法】

$$\begin{gathered} \mathrm{有}\mathrm{正}\mathrm{项}\mathrm{级}\mathrm{数}\sum _{n=1}^{\mathrm{\infty }}{u}_n,\mathrm{若}\underset{n\to \mathrm{\infty }}{lim}\frac{u_{n+1}}{u_n}=\rho \\ \mathrm{则}\mathrm{当}\rho <1，\mathrm{该}\mathrm{级}\mathrm{数}\mathrm{收}\mathrm{敛}，\mathrm{当}\rho >1，\mathrm{该}\mathrm{级}\mathrm{数}\mathrm{发}\mathrm{散}，\mathrm{当}\rho =1\mathrm{时}，\mathrm{不}\mathrm{能}\mathrm{判}\mathrm{断} \end{gathered}$$

定理 5【根值审敛法，柯西（Cauchy）审敛法】

$$\begin{gathered} \mathrm{有}\mathrm{正}\mathrm{项}\mathrm{级}\mathrm{数}\sum _{n=1}^{\mathrm{\infty }}{u}_n,\mathrm{若}\underset{n\to \mathrm{\infty }}{lim}\sqrt[n]{u_n}=\rho \\ \mathrm{则}\mathrm{当}\rho <1，\mathrm{该}\mathrm{级}\mathrm{数}\mathrm{收}\mathrm{敛}，\mathrm{当}\rho >1，\mathrm{该}\mathrm{级}\mathrm{数}\mathrm{发}\mathrm{散}，\mathrm{当}\rho =1\mathrm{时}，\mathrm{不}\mathrm{能}\mathrm{判}\mathrm{断} \end{gathered}$$

定理 6（布莱尼茨审敛法）

$$\begin{gathered} \mathrm{若}\mathrm{交}\mathrm{错}\mathrm{级}\mathrm{数}\sum _{n=1}^{\mathrm{\infty }}(-1{)}^{n-1}{u}_n\mathrm{满}\mathrm{足}: \\ 1.\mathrm{数}\mathrm{列}\{u_n\}\mathrm{单}\mathrm{调}\mathrm{递}\mathrm{减}，\mathrm{对}\mathrm{于}\mathrm{一}\mathrm{切}\mathrm{正}\mathrm{整}\mathrm{数}n,\mathrm{都}\mathrm{有}{u}_n\ge u_{n+1} \\ 2.\underset{n\to \mathrm{\infty }}{lim}{u}_n=0 \\ \mathrm{则}\mathrm{该}\mathrm{级}\mathrm{数}\mathrm{收}\mathrm{敛}，\mathrm{且}\mathrm{其}\mathrm{和}\le u_1 \end{gathered}$$

定理 7

$$\mathrm{若}\mathrm{级}\mathrm{数}\sum _{n=1}^{\mathrm{\infty }}{u}_n\mathrm{绝}\mathrm{对}\mathrm{收}\mathrm{敛}，\mathrm{则}\mathrm{级}\mathrm{数}\sum _{n=1}^{\mathrm{\infty }}{u}_n\mathrm{必}\mathrm{收}\mathrm{敛}$$$$\begin{array}{rl}& 1\end{array}$$

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