Every convergent is nearer to the value of the whole fraction than any preceding convergent.
In the case of a recurring continued fraction which represents N, where N is an integer, if n is the number of partial quotients in the recurring cycle, and pnr/gnr the nr th convergent, then p 2 nr - Ng2nr = (- I) nr, whence, if n is odd, integral solutions of the indeterminate equation x 2 - Ny 2 = I (the so-called Pellian equation) can be found.
In this case the sum to n terms of the series is equal to the nth convergent of the fraction.
We may require to represent the infinite convergent power series ao+alx+ a2x 2 + ...
Its n th convergent is not equal to the sum to n terms of the series.