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Decimal expansion of Sum_{n>=1} log(cos(1/n)) * log(sin(1/n)).
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%I #18 Mar 18 2021 05:57:17

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

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

%N Decimal expansion of Sum_{n>=1} log(cos(1/n)) * log(sin(1/n)).

%C log(cos(1/n)) ~ -1/(2*n^2) when n -> oo, so the series log(cos(1/n)) is convergent (A336603), but

%C log(sin(1/n)) ~ -log(n) when n -> oo, so the series log(sin(1/n)) is divergent.

%C However, as log(cos(1/n)) * log(sin(1/n)) ~ log(n)/(2*n^2) when n -> oo, the series log(cos(1/n)) * log(sin(1/n)) is convergent.

%D Jean-Marie Monier, Analyse, Exercices corrigés, 2ème année MP, Dunod, 1997, Exercice 3.2.1.f p. 279.

%F Equals Sum_{n>=1} log(cos(1/n)) * log(sin(1/n)).

%e 0.588251433968163564741783117942531437228475727762559...

%o (PARI) sumpos(n=1, log(cos(1/n)) * log(sin(1/n))) \\ _Michel Marcus_, Mar 18 2021

%Y Cf. A336405, A336603.

%K nonn,cons,more

%O 0,1

%A _Bernard Schott_, Mar 17 2021

%E a(4)-a(51) from _Jon E. Schoenfield_, Mar 18 2021