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) - ...

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

0,1

Comments

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.

References

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

Links

Table of n, a(n) for n=0..97.

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

Formula

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)).

Example

0.517100379042401725064810772131357...

Maple

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

Prog

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

Crossrefs

Cf. A081971, A082681.

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

Sequence in context: A275490 A052345 A197733 * A354052 A241018 A348500

Adjacent sequences: A353871 A353872 A353873 * A353875 A353876 A353877

Keyword

nonn,cons

Author

Bernard Schott, May 09 2022

Status

approved