A092267 Values 2m_0+1 = 1, 2m_1, 2m_2+1, ... associated with divergent series T shown below.

1, 454, 45891, 547208496, 3013267310449, 1961694770407970734, 589785633779065944213245, 20963601300674244910397534828794, 344117353602393170461608383214200982125

Offset

0,2

Comments

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.

References

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

Links

Table of n, a(n) for n=0..8.

Example

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

Crossrefs

Cf. A092324 (essentially the same), A002387, A056053, A092318, A092317, A092315.

Cf. A092273.

Sequence in context: A234771 A297658 A219006 * A232758 A284430 A206338

Adjacent sequences: A092264 A092265 A092266 * A092268 A092269 A092270

Keyword

nonn,nice

Author

N. J. A. Sloane, Feb 16 2004

Extensions

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

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

Status

approved