Editing Prouhet–Thue–Morse constant (section)

==Other representations==

The Prouhet–Thue–Morse constant can also be expressed, without using {{math|''t<sub>n</sub>''}} , as an infinite product,<ref name="mw">{{Mathworld|Thue-MorseConstant|Thue-Morse Constant}}</ref>
:<math> \tau = \frac{1}{4}\left[2-\prod_{n=0}^{\infty}\left(1-\frac{1}{2^{2^n}}\right)\right] </math>

This formula is obtained by substituting ''x'' = 1/2 into generating series for {{math|''t<sub>n</sub>''}}
:<math> F(x) = \sum_{n=0}^{\infty} (-1)^{t_n} x^n = \prod_{n=0}^{\infty} ( 1 - x^{2^n} ) </math>

The [[simple continued fraction|continued fraction expansion]] of the constant is [0; 2, 2, 2, 1, 4, 3, 5, 2, 1, 4, 2, 1, 5, 44, 1, 4, 1, 2, 4, 1, …] {{OEIS|A014572}}

Yann Bugeaud and Martine Queffélec showed that infinitely many partial quotients of this continued fraction are 4 or 5, and infinitely many partial quotients are greater than or equal to 50.<ref>{{cite journal |last1=Bugeaud |first1=Yann |last2=Queffélec |first2=Martine |title=On Rational Approximation of the Binary Thue-Morse-Mahler Number |journal=Journal of Integer Sequences |date=2013 |volume=16 |issue=13.2.3 |url=https://cs.uwaterloo.ca/journals/JIS/VOL16/Bugeaud/bugeaud3.html}}</ref>