Continued fraction for positive integer

It is known that

for any nonnegative integer k.

When k=0 in particular,

With some complicated induction(proof is omitted), we finally have

that is the same as above

and in general

in special form for any positive integer k.

This holds for small positive integers. For large integer, it may deviate a lot.

If we use the nearest integer k’ to k, we can calculate decimal as well. That is,

in special form

where k’ is the nearest integer to k.

We here expand k’ and adjust it by multiplying k/k’ to the first numerator.