|
|
A336603
|
|
Decimal expansion of Sum_{n>=1} log(cos(1/n)) (negated).
|
|
2
|
|
|
9, 4, 5, 3, 6, 9, 0, 5, 4, 7, 2, 6, 3, 3, 2, 9, 3, 5, 2, 6, 6, 0, 9, 5, 2, 1, 5, 4, 0, 8, 2, 7, 0, 1, 9, 8, 1, 1, 6, 9, 9, 6, 0, 9, 2, 0, 6, 6, 0, 9, 7, 9, 8, 8, 3, 7, 2, 7, 1, 4, 7, 1, 7, 7, 7, 5, 9, 4, 1, 7, 0, 6, 3, 1, 7, 1, 9, 0, 3, 8, 6, 8, 9, 4, 2, 9, 2, 1, 4, 8, 1, 3, 8, 6, 2, 4, 0, 9, 3, 3, 8, 2, 0, 1, 9
(list;
constant;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,1
|
|
COMMENTS
|
As w(n) = log(cos(1/n)) ~ -1/(2*n^2) when n -> oo, hence the series w(n) is convergent (zeta(2)/2 ~ 0.822467033...).
|
|
REFERENCES
|
Xavier Merlin, Methodix Analyse, Ellipses, 1997, Exercice 2 p. 119.
|
|
LINKS
|
|
|
FORMULA
|
Equals Sum_{n>=1} log(cos(1/n)) (negated).
Equals Sum_{k>=1} (-1)^k*2^(2*k-1)*(2^(2*k)-1)*B(2*k)*zeta(2*k)/(k*(2*k)!), where B(k) is the k-th Bernoulli number.
Equals -Sum_{k>=1} (2^(2*k)-1)*zeta(2*k)^2/(k*Pi^(2*k)). (End)
|
|
EXAMPLE
|
-0.945369054726332935266095215408270198116996...
|
|
MAPLE
|
evalf(sum(log(cos(1/n)), n=1..infinity), 50);
|
|
PROG
|
|
|
CROSSREFS
|
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|