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

Original entry on oeis.org

4, 56, 91784, 68401062
Offset: 1

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Author

Hugo Pfoertner, Feb 18 2004

Keywords

Examples

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

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

Original entry on oeis.org

1, 454, 45891, 547208496, 3013267310449, 1961694770407970734, 589785633779065944213245, 20963601300674244910397534828794, 344117353602393170461608383214200982125
Offset: 0

Views

Author

N. J. A. Sloane, Feb 16 2004

Keywords

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
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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