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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?

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    $\begingroup$ 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. $\endgroup$ – Torsten Schoeneberg Jun 22 at 4:31
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    $\begingroup$ Perhaps related to en.wikipedia.org/wiki/Gauss%27s_continued_fraction $\endgroup$ – Gerald Edgar Jun 22 at 9:17
  • $\begingroup$ Agreed @TorstenSchoeneberg $\endgroup$ – Marcus Jun 23 at 19:23
  • $\begingroup$ @GeraldEdgar: There could indeed be a connection, but I can't see it, do you? $\endgroup$ – Marcus Jun 23 at 19:25
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    $\begingroup$ crossposted (with an answer): mathoverflow.net/q/334869/11260 $\endgroup$ – Carlo Beenakker Jun 26 at 20:19

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