无穷级数，数列极限，微分学常用展开式

  1. $$
    \ln (1 + x) = \sum\limits_{n = 1}^\infty { { {( - 1)}^{n - 1} } \cdot \frac{ { {x^n} } } {n} } {\rm{ } },{\rm{ } } - 1 < x \le 1
    $$
  2. $$
    \frac{1}{2}\ln (1 + x) = \sum\limits_{n = 1}^\infty { { {( - 1)}^{n - 1} } \cdot \frac{ { {x^n} } }{2n} } {\rm{ } },{\rm{ } } - 1 < x \le 1
    $$
  3. $$
    \arctan x = \sum\limits_{n = 0}^\infty { { {( - 1)}^n} \cdot \frac{ { {x^{2n + 1} } } } { {2n + 1} } } {\rm{ , } } - 1 \le x \le 1
    $$
  4. $$
    {e^x} = \sum\limits_{n = 0}^\infty {\frac{ { {x^n} } } { {n!} }{\rm{ , } } - \infty < x < + \infty }
    $$
  5. $$
    \frac{ { {e^x} + {e^{ - x} } } }{2} = \sum\limits_{n = 0}^\infty {\frac{ { {x^{2n} } } } { {(2n)!} }{\rm{ , } } - \infty < x < + \infty }
    $$
  6. $$
    \cos x = \sum\limits_{n = 0}^\infty { { {( - 1)}^n} \cdot \frac{ { {x^{2n} } } } { {(2n)!} }{\rm{ , } } - \infty < x < + \infty }
    $$
  7. $$
    \frac{ { {e^x} - {e^{ - x} } } }{2} = \sum\limits_{n = 0}^\infty {\frac{ { {x^{2n + 1 } } } } { {(2n + 1)!} }{\rm{ , } } - \infty < x < + \infty }
    $$
  8. $$
    \sin x = \sum\limits_{n = 0}^\infty { { {( - 1)}^n} \cdot \frac{ { {x^{2n + 1} } } } { {(2n + 1)!} }{\rm{ , } } } - \infty < x < + \infty
    $$
  9. $$
    \frac{1}{ {1 + x} } = \sum\limits_{n = 0}^\infty { { {( - 1)}^n}{x^n}{\rm{ , } } } -1 < x < 1
    $$
  10. $$
    \frac{1}{ {1 - x} } = \sum\limits_{n = 0}^\infty { {x^n}{\rm{ , } } } - 1 < x < 1
    $$
  11. $$
    {(1 + x)^a} = 1 + ax + \frac{ {a(a - 1)} }{ {2!} }{x^2} + \cdots + \frac{ {a(a - 1) \cdots (a - n + 1)} }{ {n!} }{x^n} + \cdots {\rm{ , } }\left{ \begin{array}{l} x \in ( - 1,1){\rm{ , } }a \le - 1,\ x \in ( - 1,1]{\rm{ , } } - 1 < a < 0,\ x \in [ - 1,1]{\rm{ , } }a > 0.\end{array} \right.
    $$
  12. $$
    \tan x = x + \frac{1}{3}{x^3} + o({x^3})
    $$
  13. $$
    \arcsin x = x + \frac{1}{6}{x^3} + o({x^3})
    $$