# Entry in Gauss' Mathematisches Tagebuch (Mathematical Diary)

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 $$\frac{1}{1+\frac{2}{1+\frac{2}{1+\frac{8}{1+\frac{12}{1+\frac{32}{1+\frac{56}{1+128}}}}}}}$$

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.

Can anyone help or clarify?

• Do we agree that the intended continuation of the series is $\sum_{n=0}^\infty (-1)^n \cdot2^{\frac12 n(n+1)}$? (The triangular numbers seem to have occupied Gauß' mind a lot.) The sum converges $2$-adically, but not to a rational number, and I have not heard of Gauß preconceiving $p$-adic numbers. – Torsten Schoeneberg Jun 22 '19 at 4:31
• Perhaps related to en.wikipedia.org/wiki/Gauss%27s_continued_fraction – Gerald Edgar Jun 22 '19 at 9:17
• Agreed @TorstenSchoeneberg – Marcus Jun 23 '19 at 19:23
• @GeraldEdgar: There could indeed be a connection, but I can't see it, do you? – Marcus Jun 23 '19 at 19:25
• crossposted (with an answer): mathoverflow.net/q/334869/11260 – Carlo Beenakker Jun 26 '19 at 20:19