Here the fraction converges to the sum to infinity of the series. Its nth convergent is not equal to the sum to n terms of the series. Expressions for β0, β1, β2, ... by means of determinants have been given by T. Muir (Edinburgh Transactions, vol. xxvii.).
A method was given by J. H. Lambert for expressing as a continued fraction of the preceding type the quotient of two convergent power series. It is practically identical with that of finding the greatest common measure of two polynomials. As an instance leading to results of some importance consider the series
| F(n,x) = 1 + | x | + | x² | + ... |
| (γ + n)1! | (γ + n)(γ + n + 1)2! |
We have
| F(n + 1,x) - F(n,x) = - | x | F(n + 2,x), |
| (γ + n)(γ + n + 1)2! |
whence we obtain
| F(1,x) | = | 1 | x ⁄ γ(γ + 1) | x ⁄ (γ + 1)(γ + 2) | |||
| F(0,x) | 1 | + | 1 | + | 1 | + ..., |
which may also be written
| γ | x | x | |||
| γ | + | γ + 1 | + | γ + 2 | + ... |
By putting ± x² ⁄ 4 for x in F(0,x) and F(1,x), and putting at the same time γ = 1 ⁄ 2, we obtain