The terminating continued fractions
| a1, a1 + | b2 | , a1 + | b2 | b3 | , a1 + | b2 | b3 | b4 | , ... | ||||
| a2 | a2 | + | a3 | a2 | + | a3 | + | a4 |
reduced to the forms
| a1 | , | a1a2 + b2 | , | a1a2a3 + b2a3 + b2a1 | , | a1a2a3a4 + b2a3a4 + b3a1a4 + b4a1a2 + b2b4 | , ... |
| 1 | a2 | a2a3 + b3 | a2a3a4 + a4b3 + a2b4 |
are called the successive convergents to the general continued fraction.
Their numerators are denoted by p1, p2, p3, p4...; their denominators by q1, q2, q3, q4....
We have the relations
pn = anpn-1 + bnpn-2, qn = anqn-1 + bnqn-2.
In the case of the fraction
| a1 - | b2 | b3 | b4 | ..., | |||
| a2 | - | a3 | - | a4 | - |