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A121225 Decimal expansion of -log(2-2*cos(1))/2. 5
4, 2, 0, 1, 9, 5, 0, 5, 8, 2, 5, 3, 6, 8, 9, 6, 1, 7, 2, 5, 7, 9, 8, 3, 8, 4, 0, 3, 7, 9, 0, 2, 0, 3, 7, 1, 2, 4, 5, 3, 8, 9, 2, 0, 5, 5, 7, 0, 3, 4, 4, 1, 7, 6, 9, 9, 5, 6, 8, 8, 8, 9, 9, 6, 8, 5, 6, 8, 9, 8, 9, 9, 1, 5, 7, 2, 4, 7, 7, 1, 3, 4, 1, 1, 4, 6, 2, 9, 4, 7, 2, 7, 4, 6, 8, 4, 4, 6, 0, 4 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET
-1,1
COMMENTS
From Bernard Schott, Apr 20 2021: (Start)
The series Sum_{k>=1} cos(k)/k and also Sum_{k>=1} sin(k)/k (A096444) are called Fresnel series.
Abel summation shows these two series are convergent.
The series Sum_{k>=1} |cos(k)/k| is divergent. (End)
REFERENCES
Xavier Merlin, Methodix Analyse, Ellipses, 1997, p. 117.
LINKS
Université de Rennes, Séries semi-convergentes, Base raisonnée d'exercices de Mathématiques, p. 2 (Braise).
Wikipedia, Summation by parts.
FORMULA
Equals Sum_{k>=1} cos(k)/k.
Equals -log(2*sin(1/2)). - Jianing Song, Nov 09 2019
Equals log(csc(1/2)/2). - Peter Luschny, Apr 04 2020
EXAMPLE
0.0420195058253689...
MATHEMATICA
RealDigits[N[ -Log[2 - 2 Cos[1]]/2, 101]]
PROG
(PARI) -log(2-2*cos(1))/2 \\ Charles R Greathouse IV, May 15 2019
CROSSREFS
Sequence in context: A093556 A021242 A088393 * A216715 A049430 A055356
KEYWORD
cons,nonn
AUTHOR
Fredrik Johansson, Aug 20 2006
STATUS
approved

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Last modified April 25 01:35 EDT 2024. Contains 371964 sequences. (Running on oeis4.)