  We consider Jacobi matrices $J$ whose parameters have the power asymptotics
$\rho_n=n^{\beta_1} \left( x_0 + \frac{x_1}{n} + {\rm
O}(n^{-1-\epsilon})\right)$ and $q_n=n^{\beta_2} \left( y_0 + \frac{y_1}{n} +
{\rm O}(n^{-1-\epsilon})\right)$ for the off-diagonal and diagonal,
respectively. We show that for $\beta_1 > \beta_2$, or $\beta_1=\beta_2$ and
$2x_0 > |y_0|$, the matrix $J$ is in the limit circle case and the convergence
exponent of its spectrum is $1/\beta_1$. Moreover, we obtain upper and lower
bounds for the upper density of the spectrum. When the parameters of the matrix
$J$ have a power asymptotic with one more term, we characterise the occurrence
of the limit circle case completely (including the exceptional case $\lim_{n\to
\infty} |q_n|\big/ \rho_n = 2$) and determine the convergence exponent in
almost all cases.
