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)
“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)
“A joke, even if it be a lame one, is nowhere so keenly relished or quickly applauded as in a murder trial.”
—Mark Twain [Samuel Langhorne Clemens] (1835–1910)
“Growing up means letting go of the dearest megalomaniacal dreams of our childhood. Growing up means knowing they can’t be fulfilled. Growing up means gaining the wisdom and skills to get what we want within the limitations imposed by reality—a reality which consists of diminished powers, restricted freedoms and, with the people we love, imperfect connections.”
—Judith Viorst (20th century)
“Galileo, with an operaglass, discovered a more splendid series of celestial phenomena than anyone since.”
—Ralph Waldo Emerson (1803–1882)
“Nature has not placed us in an inferior rank to men, no more than the females of other animals, where we see no distinction of capacity, though I am persuaded if there was a commonwealth of rational horses ... it would be an established maxim amongst them that a mare could not be taught to pace.”
—Mary Wortley, Lady Montagu (1689–1762)
“At Timon’s villalet us pass a day,
Where all cry out,What sums are thrown away!’”
—Alexander Pope (1688–1744)