Brjuno number

In mathematics, a Brjuno number is a special type of irrational number. 
Formal definition.
An irrational number formula_1 is called a Brjuno number when the infinite sum 
converges to a finite number
Here:
Name.
The Brjuno numbers are named after Alexander Bruno, who introduced them in ; they are also occasionally spelled Bruno numbers or Bryuno numbers.
Importance.
The Brjuno numbers are important in the one–dimensional analytic small divisors problems. Bruno improved the diophantine condition in Siegel's Theorem, showed that germs of holomorphic functions with linear part formula_7 are linearizable if formula_1 is a Brjuno number. showed in 1987 that this condition is also necessary, and for quadratic polynomials is necessary and sufficient.
Properties.
Intuitively, these numbers do not have many large "jumps" in the sequence of convergents, in which the denominator of the ("n" + 1)th convergent is exponentially larger than that of the "n"th convergent. Thus, in contrast to the Liouville numbers, they do not have unusually accurate diophantine approximations by rational numbers.
Brjuno function.
The real Brjuno function formula_9 is defined for irrational "x" and satisfies