Conversion from Taylor expansion to continued fraction

Taylor expansion around zero is called MacLaurin expansion. That is,

or

since 1!=1 and x^0=1. Some expansions start from n=1. We can either calculate this expansion or find expansion formula in any mathematical book or Web site.

The easiest example is MacLaurin expansion of exp(x). Since f(0)=f’(0)=f’’(0)= . . . =1when f(x)=exp(x),

Now we use the formula from series to continued fraction in 0216. That is,

  (regular expression)

   in special form

Applying gamma-shaped multipliers repeatedly,

Thus

   in special form

When x=1

                        in special form