Pics of CFs

\phi

    \[\phi=1+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{1+\dots}}}}\]

phi proper
phi


    \[\phi=2-\cfrac{1}{2+\cfrac{1}{2-\cfrac{1}{2+\cfrac{1}{2-\dots}}}}\]

phi 2 -1 2 1 2 -1 2 1 2 -1 2 etc


    \[\phi=2-\cfrac{1}{3-\cfrac{1}{3-\cfrac{1}{3-\cfrac{1}{3-\dots}}}}\]

2 -1 3 -1 3 -1 3 -1 3 -1 3 -1 3 -1 3 phi


    \[\phi=1+\cfrac{1}{1+\cfrac{2}{2+\cfrac{2}{1+\cfrac{1}{1+\cfrac{1}{1+\dots}}}}}\]

1-11-22-21-11-11etc phi


    \[\phi=1+\cfrac{2}{2+\cfrac{2}{1+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{1+\dots}}}}}\]

phi v2


    \[\phi=1+\cfrac{3}{3+\cfrac{3}{1+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{1+\dots}}}}}\]

phi v3


\pi

    \[\pi\approx \cfrac{4}{1+\cfrac{1}{3+\cfrac{4}{5+\cfrac{9}{7}}}}\]

pi 0-41-13-45-97 etc


    \[\pi\approx \cfrac{4}{1+\cfrac{1}{3+\cfrac{4}{5+\cfrac{9}{9}}}}\]

pi approx


e

    \[e-1=1+\cfrac{1}{1+\cfrac{1}{2+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{4+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{6+\dots}}}}}}}}\]

e-1


    \[\sqrt{2}=1+\cfrac{1}{2+\cfrac{1}{2+\cfrac{1}{2+\cfrac{1}{2+\dots}}}}\]

root 2 - 1-12-12-12-12etc


    \[\sqrt{2}=\cfrac{2}{1+\cfrac{1}{1+\cfrac{2}{1+\cfrac{1}{1+\dots}}}}\]

root 2


    \[\sqrt{3}=1+\cfrac{1}{1+\cfrac{1}{2+\cfrac{1}{1+\cfrac{1}{2+\dots}}}}\]

root3


    \[\gamma=\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{2+\cfrac{1}{1+\cfrac{1}{2+\cfrac{1}{1+\cfrac{1}{4+\cfrac{1}{3+\cfrac{1}{1+\cfrac{1}{13+\dots}}}}}}}}}}\]

gamma


    \[\zeta(3)=1+\cfrac{1}{4+\cfrac{1}{1+\cfrac{1}{18+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{4+\cfrac{1}{9+\cfrac{1}{9+\dots}}}}}}}}}=1.2020569\dots\]

zeta 3


    \[\frac{-2+\sqrt{7}}{3}=1-\cfrac{2}{3-\cfrac{1}{3-\cfrac{2}{3-\cfrac{1}{3-\dots}}}}\]

1 -2 3 -1 3 -2 3 -1 3 -2 3 etc - -2pr7 o3


    \[1=\cfrac{2}{1+\cfrac{2}{1+\cfrac{2}{1+\cfrac{2}{1+\dots}}}}\]

1


    \[1=\cfrac{1}{2-\cfrac{2}{3-\cfrac{2}{3-\cfrac{2}{3-\cfrac{2}{3-\dots}}}}}\]

approx 1


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