A342647 Decimal expansion of Sum_{n>=1} log(cos(1/n)) * log(sin(1/n)).

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

Offset

0,1

Comments

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

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

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.

References

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

Links

Table of n, a(n) for n=0..50.

Formula

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

Example

0.588251433968163564741783117942531437228475727762559...

Prog

(PARI) sumpos(n=1, log(cos(1/n)) * log(sin(1/n))) \\ Michel Marcus, Mar 18 2021

Crossrefs

Cf. A336405, A336603.

Sequence in context: A231786 A007450 A303816 * A200297 A155735 A153611

Adjacent sequences: A342644 A342645 A342646 * A342648 A342649 A342650

Keyword

nonn,cons,more

Author

Bernard Schott, Mar 17 2021

Extensions

a(4)-a(51) from Jon E. Schoenfield, Mar 18 2021

Status

approved