Uniqueness of Quickly Growing Series With Rational Sums
As Sylvester himself observed, Sylvester's sequence seems to be unique in having such quickly growing values, while simultaneously having a series of reciprocals that converges to a rational number.
To make this more precise, it follows from results of Badea (1993) that, if a sequence of integers grows quickly enough that
and if the series
converges to a rational number A, then, for all n after some point, this sequence must be defined by the same recurrence
that can be used to define Sylvester's sequence.
Erdős (1980) conjectured that, in results of this type, the inequality bounding the growth of the sequence could be replaced by a weaker condition,
Badea (1995) surveys progress related to this conjecture; see also Brown (1979).
Read more about this topic: Sylvester's Sequence
Famous quotes containing the words uniqueness of, uniqueness, quickly, growing, series, rational and/or sums:
“Until now when we have started to talk about the uniqueness of America we have almost always ended by comparing ourselves to Europe. Toward her we have felt all the attraction and repulsions of Oedipus.”
—Daniel J. Boorstin (b. 1914)
“Until now when we have started to talk about the uniqueness of America we have almost always ended by comparing ourselves to Europe. Toward her we have felt all the attraction and repulsions of Oedipus.”
—Daniel J. Boorstin (b. 1914)
“Late hours, nocturnal cigars, and midnight drinkings, pleasurable though they may be, consume too quickly the free-flowing lamps of youth, and are fatal at once to the husbanded candle-ends of age.”
—Anthony Trollope (1815–1882)
“It is not growing like a tree
In bulk, doth make man better be,
Or standing long an oak, three hundred year,
To fall a log at last, dry, bald, and sere:
A lily of a day
Is fairer far in May
Although it fall and die that night;
It was the plant and flower of light.
In small proportions we just beauties see,
And in short measures life may perfect be.”
—Ben Jonson (1572–1637)
“I thought I never wanted to be a father. A child seemed to be a series of limitations and responsibilities that offered no reward. But when I experienced the perfection of fatherhood, the rest of the world remade itself before my eyes.”
—Kent Nerburn (20th century)
“Social and scientific progress are assured, sir, once our great system of postpossession payments is in operation, not the installment plan, no sir, but a system of small postpossession payments that clinch the investment. No possible rational human wish unfulfilled. A man with a salary of fifty dollars a week can start payments on a Rolls-Royce, the Waldorf-Astoria, or a troupe of trained seals if he so desires.”
—John Dos Passos (1896–1970)
“If God lived on earth, people would break his windows.”
—Jewish proverb, quoted in Claud Cockburn, Cockburn Sums Up, epigraph (1981)