Conversion from binomial expansion to continued fraction

Recall that

in special form as in 0225.

Applying the rule of signs from front to up, we get

and erasing each 2 to 1, we get

We know the following rule

as in 0113. Its consecutive form is

where every six terms are combined into three.

Applying this consecutive rule, we get

Applying gamma-shaped multipliers repeatedly,

Applying the rule of signs(up->front) and arranging the terms, we finally get

Substituting x=x-1, we get

Applying gamma-shaped multipliers of 1/(x-1) repeatedly, we get

When x=(x+1)/(x-1),

since