Somos' quadratic recurrence constant

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In mathematics, Somos' quadratic recurrence constant is defined as the number

\sigma = \sqrt {1 \sqrt {2 \sqrt{3 \ldots}}} = 1^{1/2}\;2^{1/4}\; 3^{1/8} \ldots

This can be easily re-written into the far more quickly converging product representation

\sigma = \sigma^2/\sigma = \left(\frac{2}{1} \right)^{1/2} \left(\frac{3}{2} \right)^{1/4} \left(\frac{4}{3} \right)^{1/8} \left(\frac{5}{4} \right)^{1/16} \ldots

Sondow gives a representation in terms of the derivative of the Lerch transcendent:

\ln \sigma = \frac{-1}{2} \frac {\partial \Phi} {\partial s} \left( \frac{1}{2}, 0, 1 \right)

where ln is the natural logarithm and Φ(z,s,q) is the Lerch transcendent.

A series representation, as a sum over the binomial coefficient, is also given:

\ln \sigma=\sum_{n=1}^\infty (-1)^n \sum_{k=0}^n (-1)^k {n \choose k} \ln (k+1)

Finally,

\sigma = 1.661687949335941212958...\;

[edit] References

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