無窮數列

出自高材生

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  • 設二數列 \left\{ a_{n} \right\},\left\{ b_{n} \right\} 都是收斂數列，則：
    1. \left\{ \left( a_{n}+b_{n} \right) \right\},\left\{ \left( a_{n}-b_{n} \right) \right\},\left\{ \left( a_{n}\times b_{n} \right) \right\} 都是收斂數列，
    2. \left( \underset{n\to \infty }{\mathop{\lim }}\,b_{n}=\beta \ne 0 \right)\Rightarrow \left\{ \left( \frac{a_{n}}{b_{n}} \right) \right\} 亦為收斂數列，且
    3. \left\{ \begin{align} & \left\{ \begin{align} & \underset{n\to \infty }{\mathop{\lim }}\,a_{n}=\alpha \\ & \underset{n\to \infty }{\mathop{\lim }}\,b_{n}=\beta \\ \end{align} \right.\Rightarrow \underset{x\to \infty }{\mathop{\lim }}\,\left( a_{n}+b_{n} \right)=\underset{x\to \infty }{\mathop{\lim }}\,a_{n}+\underset{x\to \infty }{\mathop{\lim }}\,b_{n}=\alpha +\beta \\ & \left\{ \begin{align} & \underset{n\to \infty }{\mathop{\lim }}\,a_{n}=\alpha \\ & \underset{n\to \infty }{\mathop{\lim }}\,b_{n}=\beta \\ \end{align} \right.\Rightarrow \underset{x\to \infty }{\mathop{\lim }}\,\left( a_{n}-b_{n} \right)=\underset{x\to \infty }{\mathop{\lim }}\,a_{n}-\underset{x\to \infty }{\mathop{\lim }}\,b_{n}=\alpha -\beta \\ & \left\{ \begin{align} & \underset{n\to \infty }{\mathop{\lim }}\,a_{n}=\alpha \\ & \underset{n\to \infty }{\mathop{\lim }}\,b_{n}=\beta \\ \end{align} \right.\Rightarrow \underset{x\to \infty }{\mathop{\lim }}\,\left( a_{n}\times b_{n} \right)=\underset{x\to \infty }{\mathop{\lim }}\,a_{n}\times \underset{x\to \infty }{\mathop{\lim }}\,b_{n}=\alpha \times \beta \\ & \left\{ \begin{align} & \underset{n\to \infty }{\mathop{\lim }}\,a_{n}=\alpha \\ & \underset{n\to \infty }{\mathop{\lim }}\,b_{n}=\beta \ne 0 \\ \end{align} \right.\Rightarrow \underset{x\to \infty }{\mathop{\lim }}\,\left( \frac{a_{n}}{b_{n}} \right)=\frac{\underset{x\to \infty }{\mathop{\lim }}\,a_{n}}{\underset{x\to \infty }{\mathop{\lim }}\,b_{n}}=\frac{\alpha }{\beta } \\ & \left\{ \begin{align} & \underset{n\to \infty }{\mathop{\lim }}\,a_{n}=\alpha \\ & c\to \text{a constant} \\ \end{align} \right.\Rightarrow \underset{x\to \infty }{\mathop{\lim }}\,\left( c\times a_{n} \right)=c\times \underset{x\to \infty }{\mathop{\lim }}\,a_{n}=c\times \alpha \\ \end{align} \right.

取自"http://www.come2pass.com/wiki/index.php/%E7%84%A1%E7%AA%AE%E6%95%B8%E5%88%97"