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)
“A work in progress quickly becomes feral. It reverts to a wild state overnight. It is barely domesticated, a mustang on which you one day fastened a halter, but which now you cannot catch. It is a lion you cage in your study. As the work grows, it gets harder to control; it is a lion growing in strength. You must visit it every day and reassert your mastery over it. If you skip a day, you are, quite rightly, afraid to open the door to its room.”
—Annie Dillard (b. 1945)
“Hardly ever can a youth transferred to the society of his betters unlearn the nasality and other vices of speech bred in him by the associations of his growing years. Hardly ever, indeed, no matter how much money there be in his pocket, can he ever learn to dress like a gentleman-born. The merchants offer their wares as eagerly to him as to the veriest “swell,” but he simply cannot buy the right things.”
—William James (1842–1910)
“The woman’s world ... is shown as a series of limited spaces, with the woman struggling to get free of them. The struggle is what the film is about; what is struggled against is the limited space itself. Consequently, to make its point, the film has to deny itself and suggest it was the struggle that was wrong, not the space.”
—Jeanine Basinger (b. 1936)
“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)
“At Timon’s villalet us pass a day,
Where all cry out,What sums are thrown away!’”
—Alexander Pope (1688–1744)