A262023 Decimal expansion of 3*log(2)/2.

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

Offset: 1

Author

Wolfdieter Lang, Sep 08 2015

Keywords

• nonn
• cons
• easy

Comments

This is the limit of the reordered alternating harmonic series 1 + 1/3 - 1/2 + 1/5 + 1/7 - 1/4 + 1/9 + 1/11 - 1/6 + ... + ... - ..., with partial sums given in A262031/A262022. This shows that the alternating harmonic series is conditionally convergent. For original references on such series see A262031.

Examples

			1.039720770839917964125848182187264852113250201540382881181020014240...
		

Links

• Srinivasa Ramanujan, Question 260, Journal of the Indian Mathematical Society, Vol. 3 (1911), p. 43.
• Eric Weisstein's World of Mathematics, Conditional Convergence.
• Index entries for transcendental numbers.

Crossrefs

Cf. A002162, A016631, A016655, A262031, A262022.

Programs

Magma

3*Log(2)/2; // Vincenzo Librandi, Jan 01 2025

Mathematica

First@ RealDigits@ N[3 Log[2]/2, 120] (* Michael De Vlieger, Jul 26 2016 *)

PARI

3*log(2)/2 \\ Michel Marcus, Sep 13 2015

Formula

Equals 3*A002162/2.

Equals A016631/2.

3*log(2)/2 = (3/2)*Sum_{n>=1} (-1)^(n+1)/n = Sum_{n>=1} ((-1)^(n+1)/n + (-1)^(n+1)/(2*n)) = A002162 + (A016655/10). - Terry D. Grant, Jul 24 2016

Equals 1 + Sum_{k>=1} 2/((4*k)^3 - 4*k) (Ramanujan, 1911). - Amiram Eldar, Jan 01 2025