A339799 - OEIS

A339799

Decimal expansion of Sum_{m>=1} (-1)^floor(sqrt(m)) / m.

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

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OFFSET

1,2

COMMENTS

When grouped by negative and positive packs = - (1+1/2+1/3) + (1/4+1/5+1/6+1/7+1/8) - (1/9+...+1/15) + (1/16+...+1/24) +...+ (-1)^k (1/k^2 +...+ 1/((k+1)^2-1)) + ...

Sum_{m>=1} (-1)^floor(sqrt(m)) / m^q is convergent iff q > 1/2.

REFERENCES

Jean-Marie Monier, Analyse, Exercices corrigés, 2ème année MP, Dunod, 1997, Exercice 3.3.35, p. 287.

E. Ramis , C. Deschamps, J. Odoux, Analyse 2, Exercices avec solutions, Classes Préparatoires aux Grandes Ecoles Scientifiques, Masson, Paris, 1985, Exercice 1. 1.14, pp. 12-13.

LINKS

Table of n, a(n) for n=1..105.

Wikipedia, Digamma function.

FORMULA

Equals Sum_{m>=1} (-1)^floor(sqrt(m)) / m.

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

Equals Sum_{m>=1} (-1)^m * (digamma((m+1)^2) - digamma(m^2)).

EXAMPLE

-1.2940812218830910763038217183567312505011225953992043022765923395275517127938...

MAPLE

evalf(Sum((-1)^n*(Psi(n^2 + 2*n + 1) - Psi(n^2)), n = 1 .. infinity), 120); # Vaclav Kotesovec, Dec 18 2020

PROG