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)
“Suspicion is not less an enemy to virtue than to happiness; he that is already corrupt is naturally suspicious, and he that becomes suspicious will quickly be corrupt.”
—Joseph Addison (1672–1719)
“You will perceive by my preaching that I am growing old: it is the privilege of years, and I am sure you will pardon it from the purity of it’s motives.”
—Thomas Jefferson (1743–1826)
“In the order of literature, as in others, there is no act that is not the coronation of an infinite series of causes and the source of an infinite series of effects.”
—Jorge Luis Borges (1899–1986)
“It is not to be forgotten that what we call rational grounds for our beliefs are often extremely irrational attempts to justify our instincts.”
—Thomas Henry Huxley (1825–95)
“At Timon’s villalet us pass a day,
Where all cry out,What sums are thrown away!’”
—Alexander Pope (1688–1744)