Conversion from Taylor expansion to continued fraction

Taylor expansion of sine-related functions

            for

      for

　         for

Here, first several Bernoulli numbers are given by B₀=(-1), B₁=1/6, B₂=1/30, B₃=1/42, B₄=1/30, B₅=5/66, B₆=691/2730,…

To convert these, we usually use the following formula

  (regular expression)

   in special form

as in 0216.

For sin(x) in particular,

So we can regard it as

Here, we can apply the following formula

  in special form

as in 0215. That is,

Applying gamma-shaped multipliers, we get

Now we consider the following formula

in 0114. That is,

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