数列极限

等差数列 \[ a_{n}=a_{1}+(n-1)d \] \[ S_{n}=\frac{n}{2}(2a_{1}+(n-1)d)=\frac{n}{2}(a_{1}+a_{n}) \] 等比数列 \[ a_{n}=a_{1}r^{n-1} \] \[ S_{n}= \left\{ \begin{gather}{} na_{1}, &r=1\\ \frac{a_{1}(1-r^n)}{1-r}, &r\neq 1 \end{gather} \right. \]

常见数列前n项和

\[ \sum^{n}_{k=1}k=1+2+3+\dots+n=\frac{n(n+1)}{2} \] \[ \sum^{n}_{k=1}k^2=1^2+2^2+3^2+\dots+n^2=\frac{n(n+1)(2n+1)}{6} \] \[ \sum^{n}_ {k=1} \frac{1}{k(k+1)}=\frac{1}{1\times2}+\frac{1}{2\times 3}+\frac{1}{3 \times 4}+\dots+\frac{1}{n(n+1)}=\frac{n}{n+1} \]

重要数列，单调递增： \[ \lim_{ n \to \infty } \left( 1+\frac{1}{n} \right)^n=e \]

算术平均值大于集合平均值 \[ \frac{a+b}{2}\geq \sqrt{ ab } \]

• 原数列收敛，子列也收敛
• 子列发散，原数列发散
• 两子列收敛到不同极限，原数列发散

\[ \lim_{ n \to \infty } a_{n}= 0 \Leftrightarrow \lim_{ n \to \infty } |a_{n}|= 0 \]

\[ \lim_{ x \to x_{0} } f(x) =A \Leftrightarrow \lim_{ x \to x_{0} } |f(x)|=|A| \] 脱帽法 \[ \lim_{ n \to \infty } x_{n}>a \Rightarrow x_{n}>a \] 戴帽法 \[ x_{n}\geq a \Rightarrow \lim_{ n \to \infty } x_{n}\geq a \]

和差化积 \[ \sin \alpha +\sin \beta=2\sin\frac{\alpha+\beta}{2}\cos \frac{\alpha-\beta}{2} \] \[ \sin \alpha -\sin \beta=2\sin\frac{\alpha-\beta}{2}\cos \frac{\alpha+\beta}{2} \] \[ \cos \alpha +\cos \beta=2\cos\frac{\alpha+\beta}{2}\cos \frac{\alpha-\beta}{2} \] \[ \cos \alpha -\cos \beta=-2\sin\frac{\alpha+\beta}{2}\sin \frac{\alpha-\beta}{2} \]

正余弦公式 \[ \sin(\alpha\pm \beta)=\sin \alpha \cos \beta \pm \cos \alpha \sin \beta \] \[ \cos(\alpha+\beta)=\cos \alpha \cos \beta \mp \sin \alpha \sin \beta \] \[ \tan(\alpha\pm \beta)=\frac{\tan \alpha \pm \tan \beta}{1\mp \tan \alpha \tan \beta} \]

海涅定理（归结原则）

\[ \exists \lim_{ x \to x_{0} } f(x)=A \Leftrightarrow \forall x_{n} \exists \lim_{ x_{n} \to x_{0} }f(x_{n})=A \]

夹逼准则

\[ n \cdot u_{\min}\leq \sum^{n}_{i=1}u_{i}\leq n\cdot u_{\max} \] \[ 1\cdot u_{\max}\leq \sum^{n}_{i=1}u_{i}\leq n\cdot u_{max} \] \[ \lim_{ n \to \infty }n^ \frac{1}{n}=e^0 =1 \]

\[ \lim_{ n \to \infty } \sqrt[n]{\sum^{m}_{i=1}a_{i}^n}=\max \{a_{m}\} \]

\[ 0<a<b \Leftrightarrow 0<\frac{1}{b} <\frac{1}{a} ,\ \lim_{ n \to \infty } (a^{-n}+b^{-n})^ \frac{1}{n}=\frac{1}{a} \]

\[ |a\pm b|\leq|a|+|b| \] \[ |a_{1}\pm a_{2}\pm a_{3}\pm\dots\pm a_{n}|\leq|a_{1}|+|a_{2}|+\dots+|a_{n}| \] \[ ||a|-|b||\leq |a-b| \] \[ \sqrt{ ab }\leq \frac{a+b}{2}\leq \sqrt{ \frac{a^2 +b^2}{2}}, \ (a,b\geq 0) \]

\[ |ab|\leq \frac{a^2+b^2}{2} \]

\[ \frac{u_{n}}{n}=u_{n} \cdot \frac{1}{n}\leq \frac{u_{n}^2 +\frac{1}{n}^2}{2}, \ u_{n} > 0 \]

\[ \sqrt[3]{ abc }\leq \frac{a+b+c}{3}\leq \sqrt{ \frac{a^2 +b^2+c^2}{3}}, \ (a,b,c\geq 0) \]

\[ a\geq b\geq 0, \ a^m\geq b^m(m>0), \ a^m\leq b^m(m<0) \]

\[ 0<a<x<b, \ 0<c<y<d \ \Rightarrow \frac{c}{b}< \frac{y}{ x} < \frac{d}{a} \]

\[ \sin x < x < \tan x \left( 0< x < \frac{\pi}{2} \right) \] \[ \sin x<x(x>0) \]

\[ x<\tan x< \frac{4}{\pi}x \] \[ \sin x> \frac{2}{\pi}x \] \[ \arctan x < x< \arcsin x \] \[ e^x\geq x+1 \] \[ \ln x < x -1 \] \[ \frac{1}{1+x}<\ln\left( 1+ \frac{1}{x} \right)< \frac{1}{x} \] \[ \frac{x}{1+x}<\ln(1+x)<x \]

压缩映射 \[ 0\leq |x_{n+1}-a|\leq k|x_{n}-a| \]

单调有界必有极限

\[ x_{x+1}-x_{n}> o r <0, \ \frac{x_{n+1}}{x_{n}}> or <1 \]

\[ f'(x)>0, \ x_{2}>x_{1}单增, \ x_{2}<x_{1}单减 \] \[ f'(x)<0, \ 不单调 \]