sums and continued fractions

    \[\frac{\pi}{4}=1-\fr{3}+\fr{5}-\fr{7}+\dots=\cfrac{1}{1+\cfrac{1^2}{2+\cfrac{3^2}{2+\cfrac{5^2}{2+\cfrac{7^2}{2+\dots}}}}}\]

Euler proved in 1748 that:

    \[\fr{A}-\fr{B}+\fr{C}-\fr{D}+\dots=\cfrac{1}{A+\cfrac{A^2}{B-A+\cfrac{B^2}{C-B+\cfrac{C^2}{D-C+\dots}}}}\]

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