Conversion from series with alternate signs to continued fraction

Recall that yesterday’s formula is

  (regular expression)

   in special form

If we have alternate signs between the terms in regular form,

because we apply Rule 1 of signs(up->front). Then we have

Applying Rule 1(up->front), we get

By Rule 2(down->front and next), we get

Skipping one term and applying Rule 2(down->front and next) again, we get

We can remove all the negative signs between terms by repeating this. Reversing the orders of denominators, we finally get

  (regular expression)

   in special form

When i={1,1,1,1, . . .} in particular

   (regular expression)

   in special form

These two forms are more frequently used in practical expansions.