I'll try to explain the routine that I was taught . . .Say i have the continued fraction: .
How do i calculate it's convergents?
The solution has a table which reads as follows
Column 1 is obviously the continued fraction.
Where do the convergents come from?
Place the numbers in a row of a horizontal chart.
The table looks like this:
The rows can be completed with this strange move . . .
This is from: .
The next comes from:
The rows are filled with an identical move . . .
And the table can be completed:
The convergents are of the form:
The convergents are:
Note that the convergents are alternately above and below
. . the exact value of: .