Hexo
0%

math_formula

发表于 更新于

数学公式渲染测试

ex=∑n=0∞xnn!=10!+x11!+x22!+x33!+⋯e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} = \frac{1}{0!} + \frac{x^1}{1!} + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots

sin⁡x=∑n=0∞(−1)nx2n+1(2n+1)!=x11!−x33!+x55!−⋯\sin x = \sum_{n=0}^{\infty} (-1)^n \frac{x^{2n+1}}{(2n+1)!} = \frac{x^1}{1!} - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots

cos⁡x=∑n=0∞(−1)nx2n(2n)!=10!−x22!+x44!−⋯\cos x = \sum_{n=0}^{\infty} (-1)^n \frac{x^{2n}}{(2n)!} = \frac{1}{0!} - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots

ln⁡(1+x)=∑n=1∞(−1)n+1xnn=x11−x22+x33−⋯\ln(1+x)=\sum_{n=1}^{\infty}(-1)^{n+1}\frac{x^n}{n}=\frac{x^1}{1}-\frac{x^2}{2}+\frac{x^3}{3}-\cdots

11−x=∑n=0∞xn=x0+x1+x2+⋯\frac{1}{1-x}=\sum_{n=0}^{\infty}x^n=x^0+x^1+x^2+\cdots

arctan⁡x=∑n=0∞(−1)nx2n+12n+1=x11−x33+x55−⋯\arctan x=\sum_{n=0}^{\infty}(-1)^n\frac{x^{2n+1}}{2n+1}=\frac{x^1}{1}-\frac{x^3}{3}+\frac{x^5}{5}-\cdots

泰勒展开公式

f(x)在k处展开：f(x)=∑n=0∞f(n)(k)n!(x−k)nf(x)在k处展开：f(x)=\sum_{n=0}^{\infty}\frac{f^{(n)}(k)}{n!}(x-k)^n

e.g.sin⁡x=∑n=0∞sin⁡(n)0n!xne.g. \sin x=\sum_{n=0}^{\infty}\frac{\sin^{(n)}0}{n!}x^n

4n+1阶导为1，4n+3阶导为-1，2n阶导为0.