Sahithyan's S1 — Mathematics
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.