*I've asked that question before on [History of Science and Mathematics](https://hsm.stackexchange.com/questions/9756/entry-in-gauss-mathematisches-tagebuch-mathematical-diary) but haven't received an answer*


Does someone have a reference or further explanation on Gauß' entry from May 24, 1796 in his mathematical diary (Mathematisches Tagebuch, full scan available via <https://gdz.sub.uni-goettingen.de/id/DE-611-HS-3382323>) on page 3 regarding the divergent series
$$1-2+8-64...$$
in relation to the continued fraction 
$$\frac{1}{\displaystyle 1+\frac{\strut 2}{\displaystyle 1+\frac{\strut 2}{\displaystyle 1+\frac{\strut 8}{\displaystyle 1+\frac{\strut 12}{\displaystyle 1+\frac{\strut 32}{\displaystyle 1+\frac{\strut 56}{\displaystyle 1+128}}}}}}}$$ 


He states also - if I read it correctly - *Transformatio seriei* which could mean *series transformation*, but I don't see how he transforms from the series to the continued fraction resp. which transformation or rule he applied.


The OEIS has an entry (<https://oeis.org/A014236>) for the sequence $2,2,8,12,32,56,128$, but I don't see the connection either.


My question: Can anyone help or clarify the relationship that Gauss' used?


Torsten Schoeneberg remarked rightfully in the original question that the term in the series are $(-1)^n\cdot 2^{\frac{1}{2}n(n+1)}$ and Gerald Edgar conjectures it might be related to [Gauss' Continued Fraction](https://en.wikipedia.org/wiki/Gauss%27s_continued_fraction).



Divergent Series & Continued Fraction (from Gauss' Mathematical Diary)
The entry from May 24, 1796 is worked out in a more general form on February 16, 1797 [reproduced below from this [scan](https://gdz.sub.uni-goettingen.de/id/DE-611-HS-3382323?tify=%7B%22pages%22:[8,9],%22panX%22:0.649,%22panY%22:0.95,%22view%22:%22toc%22,%22zoom%22:0.806%7D)]


![](https://ilorentz.org/beenakker/MO/Gauss_1797.png)


$$1-a+a^3-a^6+a^{10}+\cdots=\frac{1}{\displaystyle 1+\frac{\strut a}{\displaystyle 1+\frac{\strut a^2-a}{\displaystyle 1+\frac{\strut a^3}{\displaystyle 1+\frac{\strut a^4-a^2}{\displaystyle 1+\frac{\strut a^5}{1+\cdots}}}}}}$$


so the coefficients alternate between $a^{2n+1}$ and $a^{2n}-a^n$.


Latin text: *Amplificatio prop[ositionis] penult[imae] p[aginae] 1, scilicet $\cdots$  

Unde facile omnes series ubi exp[onentes] ser[iem] sec[undi] ordinis constituunt transformantur.*


Translation: *Expanding on the proposition 1 from the next-to-last page $\cdots$  

From here one can easily transform every series the exponents of which form a series of the second order.*


These continued fractions of series of the form 
$$1+\sum\_{n=1}^\infty a^{n(n+1)}-\sum\_{n=1}^\infty a^{n^2}=\sum\_{n=0}^\infty (-1)^n a^{n(n+1)/2}$$ 
are related to theta functions, see [chapter 29](https://books.google.nl/books?id=q8H3CwAAQBAJ&pg=PT738) of "Series and Products from the Fifteenth to the Twenty-first Century". Apparently the series originated from Jakob Bernoulli (1690), see [History of Continued Fractions and Padé Approximants.](https://books.google.nl/books?id=rxzsCAAAQBAJ&pg=PA194)


