Mathematical constant
In mathematical analysis and number theory, Somos' quadratic recurrence constant or simply Somos' constant is a constant defined as an expression of infinitely many nested square roots. It arises when studying the asymptotic behaviour of a certain sequence and also in connection to the binary representations of real numbers between zero and one. The constant named after Michael Somos. It is defined by:
formula_1
which gives a numerical value of approximately:
formula_2 (sequence in the OEIS).
Sums and products.
Somos' constant can be alternatively defined via the following infinite product:
formula_3
This can be easily rewritten into the far more quickly converging product representation
formula_4
which can then be compactly represented in infinite product form by:
formula_5
Another product representation is given by:
formula_6
Expressions for formula_7 (sequence in the OEIS) include:
formula_8
formula_9
formula_10
Integrals.
Integrals for formula_7 are given by:
formula_12
formula_13
Other formulas.
The constant formula_14 arises when studying the asymptotic behaviour of the sequence
formula_15
formula_16
with first few terms 1, 1, 2, 12, 576, 1658880, ... (sequence in the OEIS). This sequence can be shown to have asymptotic behaviour as follows:
formula_17
Guillera and Sondow give a representation in terms of the derivative of the Lerch transcendent formula_18:
formula_19
If one defines the Euler-constant function (which gives Euler's constant for formula_20) as:
formula_21
one has:
formula_22
Universality.
One may define a "continued binary expansion" for all real numbers in the set formula_23, similarly to the decimal expansion or simple continued fraction expansion. This is done by considering the unique base-2 representation for a number formula_24 which does not contain an infinite tail of 0's (for example write one half as formula_25 instead of formula_26). Then define a sequence formula_27 which gives the difference in positions of the 1's in this base-2 representation. This expansion for formula_28 is now given by:
formula_29
For example the fractional part of Pi we have:
formula_30 (sequence in the OEIS)
The first 1 occurs on position 3 after the radix point. The next 1 appears three places after the first one, the third 1 appears five places after the second one, etc. By continuing in this manner, we obtain:
formula_31 (sequence in the OEIS)
This gives a bijective map formula_32, such that for every real number formula_24 we uniquely can give:
formula_34
It can now be proven that for almost all numbers formula_24 the limit of the geometric mean of the terms formula_36 converges to Somos' constant. That is, for almost all numbers in that interval we have:
formula_37
Somos' constant is universal for the "continued binary expansion" of numbers formula_24 in the same sense that Khinchin's constant is universal for the simple continued fraction expansions of numbers formula_39.
Generalizations.
The "generalized Somos' constants" may be given by:
formula_40
for formula_41. 
The following series holds:
formula_42
We also have a connection to the Euler-constant function:
formula_43
and the following limit, where formula_44 is Euler's constant:
formula_45
References.
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