Continued Fraction Expansion

The process

• Separate the integer part
• Find the inverse of the remaining part. Result will be greated than 1.
• Repeat the process for the remaining part.

Finite expansion

Take for example.

As is finite, its continued fraction expansion is also finite. And it can be written as .

Infinite expansion

For irrational numbers, the expansion will be infinite.

For example :

Conintued fraction expansion of is .

Convergence

In the case of infinite continued fraction expansion, on each ”+” part, the expansion can be separated. Each separated part will generate a sequence of numbers, which is converging to the original number.

For example, for , the sequence will be:

Here:

• Elements with the odd index are lesser than the converging value.
• Elements with the even index are greater than the converging value.