|
|
A214508
|
|
Decimal expansion of the series limit sum_{k>=1} (-1)^(k+1) sum_{t=1..k} 1/(t^2*(k+1)^2).
|
|
2
|
|
|
1, 6, 2, 6, 5, 4, 6, 6, 7, 3, 9, 7, 4, 2, 0, 0, 8, 0, 7, 7, 5, 5, 6, 4, 5, 6, 5, 1, 7, 3, 5, 9, 1, 1, 0, 1, 1, 8, 7, 0, 6, 4, 2, 0, 8, 3, 3, 7, 6, 5, 9, 9, 2, 3, 7, 2, 6, 7, 6, 3, 0, 6, 9, 8, 3, 1, 8, 4, 3, 5, 7, 7, 2, 9, 8, 2, 1, 0, 7, 4, 9, 2, 1, 6, 7, 2, 0, 0, 7, 4, 6, 3, 7, 5, 7, 4, 9, 8, 1, 0, 6, 7, 9, 6, 9
(list;
constant;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,2
|
|
COMMENTS
|
Equals the alternating sum over (-1)^(k+1)*H_k^(2)/(k+1)^2, where H_k^(2) is the harmonic sum over inverse squares, H_k^(2) = sum_{t=1..k} 1/t^2 = 1, 5/4, 49/36, 205/144, 5269/3600,..., see A007406. The sum over H_k^(2)/(k+1)^2, over the absolute values, is Pi^4/120 = 0.811742425283353...
|
|
LINKS
|
|
|
FORMULA
|
Equals -4*A099218 +13*Pi^4/288 -7*A002117*log(2)/2+log^2(2)*(Pi^2-log^2(2))/6.
|
|
EXAMPLE
|
0.162654667397420080...
|
|
MAPLE
|
a099218 := polylog(4, 1/2) ;
-4*a099218+13*Pi^4/288-7/2*Zeta(3)*log(2)+Pi^2/6*(log(2))^2-(log(2))^4/6 ;
evalf(%) ;
|
|
MATHEMATICA
|
NSum[(-1)^(k + 1)*HarmonicNumber[k, 2]/(k + 1)^2, {k, 1, Infinity}, Method -> "AlternatingSigns", WorkingPrecision -> 110] // RealDigits[#, 10, 105] & // First (* or, from formula: *) 13*Pi^4/288 + 1/6*Pi^2*Log[2]^2 - 1/6*Log[2]*(Log[2]^3 + 21*Zeta[3]) - 4*PolyLog[4, 1/2] // RealDigits[#, 10, 105]& // First (* Jean-François Alcover, Mar 06 2013 *)
|
|
PROG
|
(PARI) 13*Pi^4/288 + 1/6*Pi^2*log(2)^2 - 1/6*log(2)*(log(2)^3 + 21*zeta(3)) - 4*polylog(4, 1/2) \\ Charles R Greathouse IV, Jul 18 2014
|
|
CROSSREFS
|
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|