# 極限 (數學)

## 普通嘅數關係

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