A081971 -id:A081971 - cp's OEIS frontend

A082681 Denominator of Sum_{i=n(n-1)/2+1..n(n+1)/2} 1/i.

1, 6, 60, 2520, 20020, 1627920, 124324200, 1694579040, 446626220040, 73706596563600, 35444732266944, 24569517992362200, 3290057629552053360, 551042196782556679200, 71028805196838414285360

Offset: 1

Dean Hickerson, Apr 10 2003

Numerator is in A081971. a(n) is a divisor of A061431(n).

a(n) = denominator(sum(i = n*(n-1)/2+1, n*(n+1)/2, 1/i)); \\ Michel Marcus, Aug 29 2013

A353874 Decimal expansion of (1/1) - (1/2+1/3) + (1/4+1/5+1/6) - (1/7+1/8+1/9+1/10) + (1/11+1/12+1/13+1/14+1/15) - ...

Original entry on oeis.org

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

Offset: 0

Bernard Schott, May 09 2022

There are n terms in the n-th group v(n), from 1 / ((n^2-n+2)/2) up to 1 / ((n^2+n)/2).

As |v(n+1)| < |v(n)|, this series is convergent according to the alternating series test.

			0.517100379042401725064810772131357...

Jean-Marie Monier, Analyse, Exercices corrigés, 2ème année, MP, Dunod, 1997, Exercice 3.3.19 pp. 285 and 303 .

user115748, Power series x^n/n with one plus, then two minuses, then three plusses, and so on, MathOverflow, Jan 2024.

Cf. A081971, A082681.

Cf. A002162, A339799 (other harmonic series with + and -).

evalf(sum(sum((-1)^(n+1)/k, k= (n^2-n+2)/2..(n^2+n)/2), n=1..infinity),100);

sumalt(n=1, (-1)^(n+1)*sum(k=(n^2-n+2)/2, (n^2+n)/2, 1/k)) \\ Michel Marcus, May 09 2022

Equals Sum_{n>=1} Sum_{k = (n^2-n+2)/2..(n^2+n)/2} (-1)^(n+1) / k.

Equals Sum_{n>=1} (-1)^(n+1) * (A081971(n)/A082681(n)).

cp's OEIS Frontend

A082681 Denominator of Sum_{i=n(n-1)/2+1..n(n+1)/2} 1/i.

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A353874 Decimal expansion of (1/1) - (1/2+1/3) + (1/4+1/5+1/6) - (1/7+1/8+1/9+1/10) + (1/11+1/12+1/13+1/14+1/15) - ...

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