Harman Patil (Editor)

Irrationality sequence

Updated on
Edit
Like
Comment
Share on FacebookTweet on TwitterShare on LinkedInShare on Reddit

In mathematics, a sequence of positive integers an is called an irrationality sequence if it has the property that for every sequence xn of positive integers, the sum of the series

Contents

∑ n = 1 ∞ 1 a n x n

exists (that is, it converges) and is an irrational number. The problem of characterizing irrationality sequences was posed by Paul Erdős and Ernst G. Straus, who originally called the property of being an irrationality sequence "Property P".

Examples

The powers of two whose exponents are powers of two, 2 2 n , form an irrationality sequence. However, although Sylvester's sequence

2, 3, 7, 43, 1807, 3263443, ...

(in which each term is one more than the product of all previous terms) also grows doubly exponentially, it does not form an irrationality sequence. For, letting x n = 1 gives

1 2 + 1 3 + 1 7 + 1 43 + ⋯ = 1 ,

a series converging to a rational number. Likewise, the factorials, n ! , do not form an irrationality sequence because the sequence x n = n + 2 leads to a series with a rational sum,

∑ n = 0 ∞ 1 ( n + 2 ) n ! = 1 2 + 1 3 + 1 8 + 1 30 + 1 144 + ⋯ = 1.

Growth rate

Any sequence an that grows at a rate such that

lim sup n log ⁡ log ⁡ a n n > log ⁡ 2

is an irrationality sequence. This includes sequences that grow at a more than doubly exponential rate as well as some doubly exponential sequences that grow more quickly than the powers of powers of two.

Every irrationality sequence must grow quickly enough that

lim n → ∞ a n 1 / n = ∞ .

However, it is not known whether there exists such a sequence in which the greatest common divisor of each pair of terms is 1 (unlike the powers of powers of two) and for which

lim n → ∞ a n 1 / 2 n < ∞ .

Analogously to irrationality sequences, Hančl (1996) has defined a transcendental sequence to be an integer sequence an such that, for every sequence xn of positive integers, the sum of the series

∑ n = 1 ∞ 1 a n x n

exists and is a transcendental number.

References

Irrationality sequence Wikipedia
(Text) CC BY-SA