Conversion from Taylor expansion to continued fraction

Taylor expansion of cosine-related functions

　　　　 for

      for

　         for

Here, E is absolute values of Euler numbers.

To convert these, we usually use the following formula

  (regular expression)

   in special form

as in 0216.

For cos(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