OFFSET
1,1
COMMENTS
Ovidiu Furdui, Limits, Series, and Fractional Part Integrals, Springer, 2013, section 3.71, p. 150.
LINKS
Ovidiu Furdui, Series Involving Products of Two Harmonic Numbers, Mathematics Magazine, Vol. 84, No. 5 (2011), pp. 371-377.
FORMULA
Equals Sum_{k>=1} (k+2)/k^3.
Equals Sum_{k>=1} H(k)*H(k+1)/(k*(k+1)), where H(k) = A001008(k)/A002805(k) is the k-th harmonic number (Furdui, 2011).
Equals Sum_{k>=1} (H(k)+1)/k^2.
Equals 1 + Sum_{k>=2} H(k)/(k-1)^2.
Equals Sum_{k>=2} (k-1)^2*(zeta(k)-1).
Equals 3 + Sum_{k>=3} (-1)^(k+1)*k^2*(zeta(k)-1).
Equals Integral_{x=0..1} log(x)*(log(x)-1)/(1-x) dx.
Equals Integral_{x>=1} log(x)*(log(x)+1)/(x*(x-1)) dx.
Equals Integral_{x>=0} x*(x+1)/(exp(x)-1) dx.
EXAMPLE
4.04904787316741500727189148966892517074892248588779...
MATHEMATICA
RealDigits[Zeta[2] + 2*Zeta[3], 10, 100][[1]]
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Amiram Eldar, Jun 10 2021
STATUS
approved