$\mathrm{with}\left(\mathrm{numtheory}\right)\:$

$\mathrm{cfrac}\left(\mathrm{\π}\right)$

$3\+\frac{1}{7\+\frac{1}{15\+\frac{1}{1\+\frac{1}{292\+\frac{1}{1\+\frac{1}{1\+\frac{1}{1\+\frac{1}{2\+\frac{1}{1\+\frac{1}{3\+\mathrm{...}}}}}}}}}}}$

$\mathrm{cfrac}\left(\mathrm{\π}\,10\,\'c\'\right)\:$

$c$

$\left[3\,\frac{22}{7}\,\frac{333}{106}\,\frac{355}{113}\,\frac{103993}{33102}\,\frac{104348}{33215}\,\frac{208341}{66317}\,\frac{312689}{99532}\,\frac{833719}{265381}\,\frac{1146408}{364913}\,\frac{4272943}{1360120}\,\mathrm{...}\right]$

$\mathrm{evalf}\left[20\right]\left(c\right)$

$\left[3.\,3.1428571428571428571\,3.1415094339622641509\,3.1415929203539823009\,3.1415926530119026041\,3.1415926539214210447\,3.1415926534674367055\,3.1415926536189366234\,3.1415926535810777712\,3.1415926535914039785\,3.1415926535893891715\,\mathrm{...}\right]$

$\mathrm{abserror} ≔ \mathrm{map}\left(z\to \mathrm{evalf}\left[20\right]\left(\mathrm{\π}-z\right)\, c_{1..10}\right)$

$\mathrm{abserror}:=\left[0.1415926535897932385\,-0.0012644892673496186\,0.0000832196275290876\,-2.667641890624{10}^{-7}\,5.778906344{10}^{-10}\,-3.316278062{10}^{-10}\,1.223565330{10}^{-10}\,-2.91433849{10}^{-11}\,8.7154673{10}^{-12}\,-1.6107400{10}^{-12}\right]$

$\mathrm{cfrac}\left(\mathrm{\π}\,10\,\'c\'\,\mathrm{centered}\right)$

$3\+\frac{1}{7\+\frac{1}{16-\frac{1}{294-\frac{1}{3-\frac{1}{4-\frac{1}{5-\frac{1}{15\+\frac{1}{3-\frac{1}{2\+\frac{1}{2\+\mathrm{...}}}}}}}}}}}$

$c$

$\left[3\,\frac{22}{7}\,\frac{355}{113}\,\frac{104348}{33215}\,\frac{312689}{99532}\,\frac{1146408}{364913}\,\frac{5419351}{1725033}\,\frac{80143857}{25510582}\,\frac{245850922}{78256779}\,\frac{411557987}{131002976}\,\frac{1068966896}{340262731}\,\mathrm{...}\right]$

$\mathrm{evalf}\left[20\right]\left(c\right)$

$\left[3.\,3.1428571428571428571\,3.1415929203539823009\,3.1415926539214210447\,3.1415926536189366234\,3.1415926535914039785\,3.1415926535898153832\,3.1415926535897926594\,3.1415926535897931603\,3.1415926535897932578\,3.1415926535897932354\,\mathrm{...}\right]$

$\mathrm{abserror} ≔ \mathrm{map}\left(z\to \mathrm{evalf}\left[20\right]\left(\mathrm{\pi }-z\right)\, c_{1..10}\right)$

$\mathrm{abserror}:=\left[0.1415926535897932385\,-0.0012644892673496186\,-2.667641890624{10}^{-7}\,-3.316278062{10}^{-10}\,-2.91433849{10}^{-11}\,-1.6107400{10}^{-12}\,-2.21447{10}^{-14}\,5.791{10}^{-16}\,7.82{10}^{-17}\,-1.93{10}^{-17}\right]$

$\mathrm{cfrac}\left(\tan \left(x\right)\right)$

$\frac{x}{1-\frac{x^2}{3-\frac{x^2}{5-\frac{x^2}{7-\frac{x^2}{9-\frac{x^2}{11-\frac{x^2}{13-\frac{x^2}{15-\frac{x^2}{17-\frac{x^2}{19\+\mathrm{...}}}}}}}}}}}$

$\mathrm{cfrac}\left(\tan \left(x\right)\,1\right)$

$\frac{x}{1\+\mathrm{...}}$

$c_1≔\mathrm{cfrac}\left(\right)$

$c_1:=x$

$\mathrm{cfrac}\left(\tan \left(x\right)\,2\right)$

$\frac{x}{1-\frac{x^2}{3\+\mathrm{...}}}$

$c_2≔\mathrm{cfrac}\left(\right)$

$c_2:=-\frac{3x}{-3\+x^2}$

$c_3≔\mathrm{cfrac}\left(\mathrm{cfrac}\left(\tan \left(x\right)\,3\right)\right)$

$c_3:=\frac{1}{3}\frac{x\left(-15\+x^2\right)}{-5\+2x^2}$

$c_4≔\mathrm{cfrac}\left(\mathrm{cfrac}\left(\tan \left(x\right)\,4\right)\right)$

$c_4:=-\frac{5x\left(-21\+2x^2\right)}{105-45x^2\+x^4}$

$c_5≔\mathrm{cfrac}\left(\mathrm{cfrac}\left(\tan \left(x\right)\,5\right)\right)$

$c_5:=\frac{1}{15}\frac{x\left(945-105x^2\+x^4\right)}{63-28x^2\+x^4}$

$$\begin{gathered} \mathbf{for} i \mathbf{from} 1 \mathbf{to} 5 \mathbf{do} \\ \mathrm{series}\left(\mathrm{O}\left(\tan \left(x\right)-c_i\right)\,x\,20\right) \\ \mathbf{end}\; \end{gathered}$$

$\mathrm{O}\left(x^3\right)$

$\mathrm{O}\left(x^5\right)$

$\mathrm{O}\left(x^7\right)$

$\mathrm{O}\left(x^9\right)$

$\mathrm{O}\left(x^{11}\right)$

$\mathrm{series}\left(\tan \left(x\right)\,x\,11\right)$

$x\+\frac{1}{3}{x}^3\+\frac{2}{15}{x}^5\+\frac{17}{315}{x}^7\+\frac{62}{2835}{x}^9\+\mathrm{O}\left(x^{11}\right)$

$s≔\mathrm{convert}\left(\,\mathrm{polynom}\right)$

$s:=x\+\frac{1}{3}{x}^3\+\frac{2}{15}{x}^5\+\frac{17}{315}{x}^7\+\frac{62}{2835}{x}^9$

$\mathrm{plot}\left(\left[\tan \left(x\right)-s\,\tan \left(x\right)-c_5\right]\,x\=-1..1\, \mathrm{legend}\=\left[\mathrm{Taylor}\,\mathrm{continued} \mathrm{fraction}\right]\,\mathrm{thickness}\=2\,\mathrm{color}\=\left[\mathrm{red}\,\mathrm{blue}\right]\right)$

$\mathrm{evalf}\left(\genfrac{}{}{0ex}{}{\tan \left(x\right)-s}{\phantom{x\=a}}|\genfrac{}{}{0ex}{}{\phantom{\mathrm{f(x)}}}{x\=1}\right)$

$0.014903316$

$\mathrm{evalf}\left(\genfrac{}{}{0ex}{}{\tan \left(x\right)-c_5}{\phantom{x\=a}}|\genfrac{}{}{0ex}{}{\phantom{\mathrm{f(x)}}}{x\=1}\right)$

$3.18{10}^{-7}$

$\mathrm{cfrac}\left(\tan \left(x\right)\,x\=\mathrm{\π}\right)$

$\frac{x-\mathrm{\π}}{1-\frac{{\left(x-\mathrm{\π}\right)}^2}{3-\frac{{\left(x-\mathrm{\π}\right)}^2}{5-\frac{{\left(x-\mathrm{\π}\right)}^2}{7-\frac{{\left(x-\mathrm{\π}\right)}^2}{9\+\mathrm{...}}}}}}$

$\mathrm{cfrac}\left(\sqrt{7}\,\mathrm{periodic}\right)$

$2\+\frac{1}{1\+\frac{1}{1\+\frac{1}{1\+\frac{1}{4\+\frac{1}{1\+\frac{1}{1\+\frac{1}{1\+\frac{1}{4\+\mathrm{...}}}}}}}}}$

$\mathrm{cfrac}\left(\sqrt{7}\,\mathrm{periodic}\,\mathrm{quotients}\right)$

$\left[\left[2\right]\,\left[1\,1\,1\,4\right]\right]$

$\mathrm{cfrac}\left(\left[\left[\right]\,\left[1\,2\,3\,4\,5\right]\right]\right)$

$\frac{1}{314}\sqrt{65029}\+\frac{195}{314}$

$\mathrm{cfrac}\left(\,\mathrm{periodic}\right)$

$1\+\frac{1}{2\+\frac{1}{3\+\frac{1}{4\+\frac{1}{5\+\frac{1}{1\+\frac{1}{2\+\frac{1}{3\+\frac{1}{4\+\frac{1}{5\+\mathrm{...}}}}}}}}}}$

$\mathrm{cfrac}\left(\,20\right)$

$1\+\frac{1}{2\+\frac{1}{3\+\frac{1}{4\+\frac{1}{5\+\frac{1}{1\+\frac{1}{2\+\frac{1}{3\+\frac{1}{4\+\frac{1}{5\+\frac{1}{1\+\frac{1}{2\+\frac{1}{3\+\frac{1}{4\+\frac{1}{5\+\frac{1}{1\+\frac{1}{2\+\frac{1}{3\+\frac{1}{4\+\frac{1}{5\+\frac{1}{1\+\mathrm{...}}}}}}}}}}}}}}}}}}}}}$