--
1 - 3 - 5 - 7 - ... 1 + 3 + 5 + 7 + ...

These results were given by Lambert, and used by him to prove that [pi]
and [pi] squared incommensurable, and also any commensurable power of e.

Gauss in his famous memoir on the hypergeometric series

F([alpha], [beta], [gamma], x) =

[alpha].[beta] [alpha]([alpha] + 1)[beta]([beta] + 1)
--------------x + -------------------------------------- x squared + ...
1.[gamma] 1.2.[gamma].([gamma] + 1)

gave the expression for F([alpha], [beta] + 1, [gamma] + 1, x) /
F([alpha], [beta], [gamma], x) as a continued fraction, from which if we
put [beta] = 0 and write [gamma] - 1 for [gamma], we get the
transformation

[alpha] [alpha]([alpha] + 1)
1 + -------x + --------------------x squared +
[gamma] [gamma]([gamma] + 1)

[alpha]([alpha] + 1)([alpha] + 2)
---------------------------------x cubed + ... =
[gamma]([gamma] + 1)([gamma] + 2)

1 [beta]1 x [beta]2 x
-- --------- --------- where
1 - 1 - 1 - ...

[alpha] ([alpha] + 1)[gamma]
[beta]1 = -------, [beta]3 = --------------------------, ...,
[gamma] ([gamma] + 1)([gamma] + 2)

([alpha] + n - 1)([gamma] + n - 2)
[beta]_{2n-1} = ------------------------------------,
([gamma] + 2n - 3)([gamma] + 2n - 2)

[gamma] - [alpha] 2([gamma] + 1 - [alpha])
[beta]2 = --------------------, [beta]4 = --------------------------,
[gamma]([gamma] + 1) ([gamma] + 2)([gamma] + 3)

n([gamma] + n - 1 - [alpha])
..., [beta]_{2n} = ------------------------------------.
([gamma] + 2n - 2)([gamma] + 2n - 1)

From this we may express several of the elementary series as continued
fractions; thus taking [alpha] = 1, [gamma] = 2, and putting x for -x,

x 1 squaredx 1 squaredx 2 squaredx 2 squaredx 3 squaredx 3 squaredx
we have log(1 + x) = -- --- --- --- --- --- ---
1 + 2 + 3 + 4 + 5 + 6 + 7 + ...

Taking [gamma] = 1, writing x/[alpha] for x and increasing [alpha]
indefinitely, we have

1 x x x x x
e^x =