u3 u_{n}
1 + -- - 1 + -- - ... - 1 + -------
u1 u2 u_{n-1}

which we can transform into

u1 u2 u1u3 u2u4 u_{n-2}u_{n}
-- ------- ------- ------- ---------------,
1 - u1 + u2 - u2 + u3 - u3 + u4 - ... - u_{n-1} + u_{n}

a result given by Euler.

2. In this case the sum to n terms of the series is equal to the n^{th}
convergent of the fraction. There is, however, a different way in which
a Series may be represented by a continued fraction. We may require to
represent the infinite convergent power series a0 + a1x + a2x squared + ... by
an infinite continued fraction of the form

[beta]0 [beta]1 x [beta]2 x [beta]3 x
------- --------- --------- ---------
1 - 1 - 1 - 1 - ...

Here the fraction converges to the sum to infinity of the series. Its
n^{th} convergent is not equal to the sum to n terms of the series.
Expressions for [beta]0, [beta]1, [beta]2, ... 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

x x squared
F(n,x) = 1 + --------------- + -------------------------------- + ...
([gamma] + n)1! ([gamma] + n)([gamma] + n + 1)2!

We have

x
F(n + 1,x) - F(n,x) = - ------------------------------ F(n + 2,x),
([gamma] + n)([gamma] + n + 1)

whence we obtain

F(1,x) 1 x/[gamma]([gamma] + 1) x/([gamma] + 1)([gamma] + 2)
------ = -- ---------------------- ----------------------------
F(0,x) 1 + 1 + 1 + ...,

which may also be written

[gamma] x x
------- ----------- -----------
[gamma] + [gamma] + 1 + [gamma] + 2 + ...

By putting +- x squared/4 for x in F(0,x) and F(1,x), and putting at the same
time [gamma] = 1/2, we obtain

x x squared x squared x squared x x squared x squared x squared
tan x = -- -- -- -- tanh x = -- -- --