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:
“Somehow we have been taught to believe that the experiences of girls and women are not important in the study and understanding of human behavior. If we know men, then we know all of humankind. These prevalent cultural attitudes totally deny the uniqueness of the female experience, limiting the development of girls and women and depriving a needy world of the gifts, talents, and resources our daughters have to offer.”
—Jeanne Elium (20th century)
“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)
“The first time many women hold their tiny babies, they are apt to feel as clumsy and incompetent as any man. The difference is that our culture tells them they’re not supposed to feel that way. Our culture assumes that they will quickly learn how to be a mother, and that assumption rubs off on most women—so they learn.”
—Pamela Patrick Novotny (20th century)
“Rearing three children is like growing a cactus, a gardenia, and a tubful of impatiens. Each needs varying amounts of water, sunlight and pruning. Were I to be absolutely fair, I would have to treat each child as if he or she were absolutely identical to the other siblings, and there would be no profit for anyone in that.”
—Phyllis Theroux, U.S. journalist. “On Being Fair,” Night Lights: Bedtime Stories for Parents in the Dark, Penguin (1987)
“Every day the fat woman dies a series of small deaths.”
—Shelley Bovey, U.S. author. Being Fat Is Not a Sin, ch. 1 (1989)
“So far as discipline is concerned, freedom means not its absence but the use of higher and more rational forms as contrasted with those that are lower or less rational.”
—Charles Horton Cooley (1864–1929)
“If God lived on earth, people would break his windows.”
—Jewish proverb, quoted in Claud Cockburn, Cockburn Sums Up, epigraph (1981)