A092267 -id:A092267 - cp's OEIS frontend

A092273 Positions where the partial sums of the series in A092324 (and A092267) changes sign.

4, 56, 91784, 68401062

Offset: 1

Hugo Pfoertner, Feb 18 2004

nonn

			The sums after the 4th, 56th and 91784th terms are respectively -4.166666666666666666666666666666666E-0002, 1.103704639283586926655126617438355E-0003 and -2.723741766763051976129695675715836E-0006.

A092324 Values m_0 = 0, m_1, m_2, ... associated with divergent series T shown below.

Original entry on oeis.org

0, 227, 22945, 273604248, 1506633655224, 980847385203985367, 294892816889532972106622, 10481800650337122455198767414397, 172058676801196585230804191607100491062

Offset: 0

N. J. A. Sloane, Feb 16 2004

T = 1
- (1/2 + 1/4 + 1/6 + ... + 1/(2m_1))
+ (1/3 + 1/5 + 1/7 + ... + 1/(2m_2+1))
- (1/(2m_1+2) + 1/(2m_1+4) + ... + 1/(2m_3)
+ (1/(2m_2+3) + 1/(2m_2+5) + ... + 1/(2m_4+1))
- (1/(2m_3+2) + 1/(2m_3+4) + ... + 1/(2m_5)
+ (1/(2m_4+3) + 1/(2m_4+5) + ... + 1/(2m_6+1))
- ...
where the partial sums of the terms from 1 through the end of rows 0, 1, ... are respectively 1, just < -2, just > 3, just < -4, just > 5, etc.
Every positive number appears exactly once as a denominator in T.
The series T is a divergent rearrangement of the conditionally convergent series Sum_{j>=1} (-1)^j/j which has the entire real number system as its set of limit points.
Comment from Hans Havermann: I calculated these with Mathematica. I used NSum[1/(2i), {i, 1, x}] for the even denominators, where I had to adjust the options to obtain maximal accuracy and N[(EulerGamma + Log[4] - 2)/2 + PolyGamma[0, 3/2 + y]/2, precision] for the odd denominators. The precision needed for the last term shown was around 45 digits.

			1 - (1/2 + 1/4 + 1/6 + ... + 1/454) = -2.002183354..., which is just less than -2; so a(1) = m_1 = 227.
1 - (1/2 + 1/4 + 1/6 + ... + 1/454) + (1/3 + 1/5 + ... + 1/45891) = 3.000021113057..., which is just greater than 3; so a(2) = m_2 = 22945.

B. R. Gelbaum and J. M. H. Olmsted, Counterexamples in Analysis, Holden-Day, San Francisco, 1964; see p. 55.

Cf. A092267 (essentially the same), A002387, A056053, A092318, A092317, A092315.

Cf. A092273.

a(2) and a(3) from Hugo Pfoertner, Feb 17 2004

a(4) onwards from Hans Havermann, Feb 18 2004

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A092273 Positions where the partial sums of the series in A092324 (and A092267) changes sign.

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A092324 Values m_0 = 0, m_1, m_2, ... associated with divergent series T shown below.

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