In mathematics, Somos' quadratic recurrence constant, named after Michael Somos, is the number
\sigma=\sqrt{1\sqrt{2\sqrt{3 … }}}=11/2 21/4 31/8 … .
This can be easily rewritten into the far more quickly converging product representation
\sigma=
\sigma2 | |
\sigma |
=\left(
2 | |
1 |
\right)1/2\left(
3 | |
2 |
\right)1/4\left(
4 | |
3 |
\right)1/8\left(
5 | |
4 |
\right)1/16 … ,
which can then be compactly represented in infinite product form by:
\sigma=
infty | |
\prod | |
k=1 |
\left(1+
1 | |
k |
1/2k | |
\right) |
.
The constant
\sigma
g0=1;gn=n
2, | |
g | |
n-1 |
n>1,
with first few terms 1, 1, 2, 12, 576, 1658880, ... . This sequence can be shown to have asymptotic behaviour as follows:
gn\sim
| |||||
n+2+O(1/n) |
.
Guillera and Sondow give a representation in terms of the derivative of the Lerch transcendent:
ln\sigma=-
1 | |
2 |
\partial\Phi | |
\partials |
\left(1/2,0,1\right)
where
ln
\Phi(z,s,q)
Finally,
\sigma=1.661687949633594121296...