極限 (數學)

極限（粵拼：gik6 haan6）係現代數學（特別係數學分析學入面）嘅基礎概念，可以分做兩方面：響數列方面，極限可以用嚟描述一個數列嘅越嚟越大嗰陣，數列入面元素嘅性質變化嘅趨勢，叫數列極限；響函數方面，極限可以描述函數嘅自變量接近某一個值嗰陣，相對應的函數值變化嘅趨勢，叫函數極限。

普通嘅數關係

每個數可以計算以有無窮大量嘅極限。

例如：

• $1=\displaystyle \sum \limits _{n=1}^{\infty }2^{-1n}=\displaystyle \sum \limits _{n=1}^{\infty }{\frac {1}{2^{n}}}={\frac {1}{2^{1}}}+{\frac {1}{2^{2}}}+{\frac {1}{2^{3}}}+{\frac {1}{2^{4}}}+\cdots$
• ${\frac {1}{2}}=\displaystyle \sum \limits _{n=1}^{\infty }3^{-1n}=\displaystyle \sum \limits _{n=1}^{\infty }{\frac {1}{3^{n}}}={\frac {1}{3^{1}}}+{\frac {1}{3^{2}}}+{\frac {1}{3^{3}}}+{\frac {1}{3^{4}}}+\cdots$
• ${\frac {1}{3}}=\displaystyle \sum \limits _{n=1}^{\infty }2^{-2n}=\displaystyle \sum \limits _{n=1}^{\infty }{\frac {1}{2^{2n}}}={\frac {1}{2^{2}}}+{\frac {1}{2^{4}}}+{\frac {1}{2^{6}}}+{\frac {1}{2^{8}}}+\cdots$
• ${\frac {1}{7}}=\displaystyle \sum \limits _{n=1}^{\infty }2^{-3n}=\displaystyle \sum \limits _{n=1}^{\infty }{\frac {1}{2^{3n}}}={\frac {1}{2^{3}}}+{\frac {1}{2^{6}}}+{\frac {1}{2^{9}}}+{\frac {1}{2^{12}}}+\cdots$
• $\ln 2=\sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{n}}=1-{\frac {1}{2}}+{\frac {1}{3}}-{\frac {1}{4}}+\cdots$
• ${\frac {\pi ^{2}}{6}}=\sum _{n=1}^{\infty }{\frac {1}{n^{2}}}=1+{\frac {1}{4}}+{\frac {1}{9}}+{\frac {1}{16}}+\cdots$