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A092324
Values m_0 = 0, m_1, m_2, ... associated with divergent series T shown below.
2
0, 227, 22945, 273604248, 1506633655224, 980847385203985367, 294892816889532972106622, 10481800650337122455198767414397, 172058676801196585230804191607100491062
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.
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.
REFERENCES
B. R. Gelbaum and J. M. H. Olmsted, Counterexamples in Analysis, Holden-Day, San Francisco, 1964; see p. 55.
EXAMPLE
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.
CROSSREFS
Cf. A092267 (essentially the same), A002387, A056053, A092318, A092317, A092315.
Cf. A092273.
Sequence in context: A349985 A209843 A115998 * A122976 A098245 A343169
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