Conversion from exponential function to continued fraction

We know that

Recall that

   in special form

in 0225.

Regarding x=1/t and y=tn, it is going to be

As , 1/t, 2/t, 3/t,. . . are all going to be zeros. So, we can say that

Conversion from natural logarithm to continued fraction

We know that

Recall that

  in special form

in 0225.

because the first one is subtracted from both sides. The first numerator is independent also in continued fraction, we can get

where the term of t is cancelled out.

As t->0,

Applying Rule 1(up->front) of signs in 0109, we get

   in special form

where all the negative signs are removed. Or, replacing x by 1+x, we get

   in special form

Applying gamma-shaped multipliers of x repeatedly,

Finally, we get

　　in special form