91c:40003 40A99
   Sándor, J.
   On the irrationality of some alternating series. (English. Romanian
   summary)
   Studia Univ. Babe\c s-Bolyai Math. 33 (1988), no. 4, 8--12.
   
   From the introduction: "The aim of this paper is to find some
   irrationality criteria for alternating series. H. F. Sandham [in Otto
   Dunkel memoriam problem book, edited by E. Howard and E. P. Starke,
   1957; per bibl.] has proven the formula $$C=\sum\sp \infty\sb {n=1}
   (-1)\sp n[\ln n/\ln 2]/n$$ where $C$ is Euler's constant and $[x]$
   denotes the integer part of the real number $x$. It is well known that
   the old problem of the irrationality of $C$ is still open and there
   are only a few results related to this problem. In the light of the
   above mentioned formula it would be interesting to prove theorems on
   the irrationality of alternating series. Theorem 1. Let $(a\sb n)$
   denote a sequence of positive integers such that $a\sb n(a\sb 1a\sb
   2\cdots a\sb {n-1})\sp 2\to\infty$ for $n \to\infty$. Then the series
   $\sum\sp \infty\sb {n=1}(-1)\sp {n-1}(1/a\sb n(a\sb 1\cdots a\sb
   {n-1})\sp 2)$ is irrational. Theorem 2. Let $(b\sb n),(a\sb n)$ be two
   sequences of positive integers having the following properties: (1)
   $n\mid a\sb 1a\sb 2\cdots a\sb n$ for every $n=1,2,3,\cdots\,$; (2)
   $b\sb {n+1}/a\sb {n+1}< b\sb n<a\sb n$ for $n=1,2,3,\cdots\,$. Then
   the series $\sum\sp \infty\sb {n=1}(-1)\sp {n-1} (b\sb k/a\sb 1a\sb
   2\cdots a\sb n)$ is irrational."
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   86i:01001 01A05 (00A99 11-03)
   Ayoub, Raymond G.(1-PAS)
   Partial triumph or total failure? 
   Math. Intelligencer 7 (1985), no. 2, 55--58.

   The author discusses incorrect or incomplete proofs of theorems by
   three leading mathematicians to show that "in the failure there may
   emerge a measure of triumph". In particular, Gauss corrected Euler's
   incomplete proof of the fundamental theorem of algebra and used one of
   Euler's ideas to give a second proof. The Stieltjes statement without
   proof that the Riemann hypothesis is true seems to have led to further
   work on the zeta function, and Appell's false proof in the 1926
   Comptes Rendus de l'Academie des Sciences that Euler's constant is
   irrational used ingenious ideas. Even in their hour of failure the
   first class mathematicians appear to have a wholesome influence on
   others.
   
                          Reviewed by E. L. Cohen
   
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