User:Superdan006/Sandbox/hw1


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


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


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


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


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


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


 * $$x_1 = x_0 - \frac{f(x_0)}{f'(x_0)} $$


 * $$ x_{root}=\sqrt{a} $$
 * $$ x^2_{root}=a $$
 * $$ x^2_{root}-a=0 $$


 * $$ f(x)=x^2-a $$
 * $$ f'(x)=2x $$


 * $$x_{n+1} = x_n - \frac{x^2_n-a}{2x_n} $$


 * $$\sqrt{x} = q_0 + \cfrac{p_1}{q_1 + \cfrac{p_2}{q_2 + \ddots\,}} $$


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


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


 * $$ P_0(x) = 1$$


 * $$ P_1(x) = x$$


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


 * $$ P_4(x) = \frac{35}{8}x^4-\frac{15}{4}x^2+\frac{3}{8} $$


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


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


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


 * $$ x_{root}=log_2(a) $$
 * $$ 2^{x_{root}}=a $$
 * $$ 2^{x_{root}}-a=0 $$


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


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