Somos' Quadratic Recurrence Constant
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High Quality Content by WIKIPEDIA articles! In mathematics, Somos' quadratic recurrence constant, named after Michael Somos, a researcher in the Georgetown University Mathematics Department, is the number sigma = sqrt {1 sqrt {2 sqrt{3 cdots}}} = 1^{1/2};2^{1/4}; 3^{1/8} cdots., 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} cdots. The constant arises when studying the asymptotic behaviour of the sequence g_0=1, ...
High Quality Content by WIKIPEDIA articles! In mathematics, Somos' quadratic recurrence constant, named after Michael Somos, a researcher in the Georgetown University Mathematics Department, is the number sigma = sqrt {1 sqrt {2 sqrt{3 cdots}}} = 1^{1/2};2^{1/4}; 3^{1/8} cdots., 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} cdots. The constant arises when studying the asymptotic behaviour of the sequence g_0=1, ; , g_n = ng_{n-1}^2, qquad n 1, , with first few terms 1, 1, 2, 12, 576, 1658880 ... (sequence A052129 in OEIS). This sequence can be shown to have asymptotic behaviour as follows: g_n sim frac {sigma^{2^n}}{n + 2 + O(frac{1}{n})}.
