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

Author

N. J. A. Sloane, Feb 16 2004

Keywords

• nonn
• nice

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.

Examples

			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.
		

References

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

Crossrefs

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

Cf. A092273.

Extensions

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

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