cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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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) - ...

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

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Author

Bernard Schott, May 09 2022

Keywords

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.

Examples

			0.517100379042401725064810772131357...
		

References

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

Crossrefs

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

Programs

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

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)).
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