| 1 | and | 1 | . |
| q2n | qn(qn + qn+1) |
Every simple continued fraction must converge to a definite limit; for its value lies between that of the first and second convergents and, since
| pn | ~ | pn-1 | = | 1 | , Lt. | pn | = Lt. | pn-1 | , |
| qn | qn-1 | qnqn-1 | qn | qn-1 |
so that its value cannot oscillate.
The chief practical use of the simple continued fraction is that by means of it we can obtain rational fractions which approximate to any quantity, and we can also estimate the error of our approximation. Thus a continued fraction equivalent to π (the ratio of the circumference to the diameter of a circle) is
| 3 + | 1 | 1 | 1 | 1 | 1 | 1 | ... | ||||||
| 7 | + | 15 | + | 1 | + | 292 | + | 1 | + | 1 | + |
of which the successive convergents are
| 3 | , | 22 | , | 333 | , | 355 | , | 103993 | , &c., |
| 1 | 7 | 106 | 113 | 33102 |
the fourth of which is accurate to the sixth decimal place, since the error lies between 1/{q4q5} or .0000002673 and a6/{q4q6} or .0000002665.
Similarly the continued fraction given by Euler as equivalent to ½(e - 1) (e being the base of Napierian logarithms), viz.