%I #22 Jan 02 2023 03:27:49

%S 7,7,9,5,1,1,5,3,7,3,9,3,2,8,1,5,6,9,2,8,4,0,1,0,7,3,8,6,8,8,8,5,7,5,

%T 7,1,5,2,2,1,4,0,0,7,3,1,1,7,5,4,1,6,5,9,4,6,3,3,3,9,6,5,0,8,2,5,5,3,

%U 3,0,6,5,8,2,0,8,4,7,9,7,5,9,6,7,1,9,1,6,3,0,8,8,0,2,7,3,8,2,1

%N Decimal expansion of Sum_{k>=1} (-1)^(k+1) / k^(1+1/k).

%C This series is convergent according to the alternating series test.

%D J.-M. Monier, Analyse, Tome 3, 2ème année, MP.PSI.PC.PT, Dunod, 1997, Exercice 3.3.9.a p. 266.

%H Mark J. Cooker, <a href="https://www.jstor.org/stable/23248679">Fast formulas for slowly convergent alternating series</a>, The Mathematical Gazette, Vol. 95, No. 533 (2011), pp. 218-226; <a href="https://www.jstor.org/stable/23248540">Correspondence</a>, ibid., pp. 560-561 (erratum).

%F Equals Sum_{n>=1} (-1)^(n+1) / n^(1+1/n).

%e 0.7795115373932815692840107386888575715...

%p evalf(sum((-1)^(n+1)/n^(1+1/n), n= 1..infinity),120);

%t RealDigits[NSum[(-1)^(k+1)/k^((k+1)/k), {k, 1, Infinity}, WorkingPrecision -> 120, Method -> "AlternatingSigns"], 10, 105][[1]] (* _Amiram Eldar_, Jan 02 2023 *)

%o (PARI) sumalt(k=1, (-1)^(k+1) / k^(1+1/k)) \\ _Michel Marcus_, May 02 2022

%Y Cf. A002162 (Sum_{k>=1} (-1)^(k+1)/k).

%K nonn,cons

%O 0,1

%A _Bernard Schott_, May 01 2022