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%I #32 Dec 19 2020 04:53:45
%S 1,2,9,4,0,8,1,2,2,1,8,8,3,0,9,1,0,7,6,3,0,3,8,2,1,7,1,8,3,5,6,7,3,1,
%T 2,5,0,5,0,1,1,2,2,5,9,5,3,9,9,2,0,4,3,0,2,2,7,6,5,9,2,3,3,9,5,2,7,5,
%U 5,1,7,1,2,7,9,3,8,5,1,5,7,1,2,0,9,0,3,6,2,6,1,8,4,8,6,1,4,2,7,8,9,6,0,8,2
%N Decimal expansion of Sum_{m>=1} (-1)^floor(sqrt(m)) / m.
%C When grouped by negative and positive packs = - (1+1/2+1/3) + (1/4+1/5+1/6+1/7+1/8) - (1/9+...+1/15) + (1/16+...+1/24) +...+ (-1)^k (1/k^2 +...+ 1/((k+1)^2-1)) + ...
%C Sum_{m>=1} (-1)^floor(sqrt(m)) / m^q is convergent iff q > 1/2.
%D Jean-Marie Monier, Analyse, Exercices corrigés, 2ème année MP, Dunod, 1997, Exercice 3.3.35, p. 287.
%D E. Ramis , C. Deschamps, J. Odoux, Analyse 2, Exercices avec solutions, Classes Préparatoires aux Grandes Ecoles Scientifiques, Masson, Paris, 1985, Exercice 1. 1.14, pp. 12-13.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Digamma_function">Digamma function</a>.
%F Equals Sum_{m>=1} (-1)^floor(sqrt(m)) / m.
%F Equals Sum_{m>=1} (-1)^m * Sum_{k=m^2..(m+1)^2-1} 1/k.
%F Equals Sum_{m>=1} (-1)^m * (digamma((m+1)^2) - digamma(m^2)).
%e -1.2940812218830910763038217183567312505011225953992043022765923395275517127938...
%p evalf(Sum((-1)^n*(Psi(n^2 + 2*n + 1) - Psi(n^2)), n = 1 .. infinity), 120); # _Vaclav Kotesovec_, Dec 18 2020
%o (PARI) sumalt(k=1, (-1)^k * (psi(1 + 2*k + k^2) - psi(k^2))) \\ _Vaclav Kotesovec_, Dec 18 2020
%Y Cf. A196521, A228639, A231902.
%K nonn,cons
%O 1,2
%A _Bernard Schott_, Dec 17 2020
%E More terms from _Vaclav Kotesovec_, Dec 18 2020
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