A345203 Decimal expansion of zeta(2) + 2*zeta(3).

4, 0, 4, 9, 0, 4, 7, 8, 7, 3, 1, 6, 7, 4, 1, 5, 0, 0, 7, 2, 7, 1, 8, 9, 1, 4, 8, 9, 6, 6, 8, 9, 2, 5, 1, 7, 0, 7, 4, 8, 9, 2, 2, 4, 8, 5, 8, 8, 7, 7, 9, 6, 2, 0, 1, 3, 2, 0, 1, 0, 1, 3, 4, 0, 0, 5, 3, 6, 8, 3, 8, 8, 1, 9, 7, 5, 8, 2, 7, 0, 5, 4, 2, 0, 6, 5, 4

Offset: 1

Author

Amiram Eldar, Jun 10 2021

Keywords

• nonn
• cons

Comments

Ovidiu Furdui, Limits, Series, and Fractional Part Integrals, Springer, 2013, section 3.71, p. 150.

Examples

			4.04904787316741500727189148966892517074892248588779...
		

Links

• Ovidiu Furdui, Series Involving Products of Two Harmonic Numbers, Mathematics Magazine, Vol. 84, No. 5 (2011), pp. 371-377.

Crossrefs

Cf. A001008, A002117, A002805, A013661, A218505, A345204.

Programs

Mathematica

RealDigits[Zeta[2] + 2*Zeta[3], 10, 100][[1]]

Formula

Equals A013661 + 2 * A002117.

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.