log(1 + x) = x 1²x 1²x 2²x 2²x 3²x 3²x
1 + 2 + 3 + 4 + 5 + 6 + 7 + ...

Taking γ = 1, writing x ⁄ α for x and increasing α indefinitely, we have

ex = 1 x x x x x
1 - 1 + 2 - 3 + 2 - 5 + ...

For some recent developments in this direction the reader may consult a paper by L. J. Rogers in the Proceedings of the London Mathematical Society (series 2, vol. 4).

Ascending Continued Fractions.

There is another type of continued fraction called the ascending continued fraction, the type so far discussed being called the descending continued fraction. It is of no interest or importance, though both Lambert and Lagrange devoted some attention to it. The notation for this type of fraction is

b4 + b5 +
b3 + a5
b2 + a4
a1 + a3
a2

It is obviously equal to the series

a1 + b2 + b3 + b4 + b5 + ...
a2a2a3a2a3a4a2a3a4a5

Historical Note.