User:Superdan006/Sandbox/hw1

$\pi =4\sum _{k=0}^{\infty }{\frac {(-1)^{k}}{2k+1}}$ (3)

${\frac {2}{\pi }}=\sum _{k=0}^{\infty }(-1)^{k}(4k+1)\prod _{k=1}^{\infty }{\frac {2k-1}{2k}}$ (4)

$f(x)=\sum _{n=0}^{\infty }{\frac {f^{(n)}(x)}{n!}}(x-a)^{n}$ (5)

$f(x)={\sqrt {x}}$ (6)

$f(x)={\sqrt {1+x}}=\sum _{n=0}^{\infty }{\frac {(-1)^{n}(2n)!}{(1-2n)(n!)^{2}}}{\frac {x^{n}}{4^{n}}}$ (7)

$f(x)={\sqrt {N^{2}+x}}=\sum _{n=0}^{\infty }{\frac {(-1)^{n}(2n)!}{(1-2n)(n!)^{2}}}{\frac {x^{n}}{4^{n}N^{2n-1}}}$ (7)

$x_{1}=x_{0}-{\frac {f(x_{0})}{f'(x_{0})}}$

$x_{root}={\sqrt {a}}$

$x_{root}^{2}=a$

$x_{root}^{2}-a=0$

$f(x)=x^{2}-a$

$f'(x)=2x$

$x_{n+1}=x_{n}-{\frac {x_{n}^{2}-a}{2x_{n}}}$

${\sqrt {x}}=q_{0}+{\cfrac {p_{1}}{q_{1}+{\cfrac {p_{2}}{q_{2}+\ddots \,}}}}$

${\sqrt {x}}=a+{\cfrac {b}{2a+{\cfrac {b}{2a+\ddots \,}}}}$

${d \over dx}\left[(1-x^{2}){d \over dx}P_{n}(x)\right]+n(n+1)P_{n}(x)=0$

$P_{0}(x)=1$

$P_{1}(x)=x$

$P_{n+1}(x)={\frac {1}{n+1}}xP_{n}(x)-nP_{n-1}(x)$

$P_{4}(x)={\frac {35}{8}}x^{4}-{\frac {15}{4}}x^{2}+{\frac {3}{8}}$

$x^{2}{\frac {d^{2}y}{dx^{2}}}+x{\frac {dy}{dx}}+(x^{2}-\alpha ^{2})y=0$

$J_{n-1}(x)={\frac {2n}{x}}J_{n}(x)-J_{n+1}(x)$

$J_{0}(x)+2\sum _{n=1}^{\infty }J_{2n}(x)=1$

$x_{root}=log_{2}(a)$

$2^{x_{root}}=a$

$2^{x_{root}}-a=0$

$f(x)=2^{x}-a$

$f'(x)=ln(2)2^{x}$

$x_{n+1}=x_{n}-{\frac {1}{ln(2)}}+{\frac {a}{ln(2)2^{x_{n}}}}$