This critical value is $\alpha_0 \approx 1.3056867$ ... For $\alpha > \alpha_0$ the series converges, for $\alpha < \alpha_0$ it diverges.

Before the inexpensive computer, it was difficult to tell whether the critical value is ${}> 1$ or not.

Boyd, David W. The sequence of radii of the Apollonian packing. Math. Comp. 39 (1982), no. 159, 249–254.

http://www.ams.org/mathscinet-getitem?mr=658230

This critical value is $\alpha_0 \approx 1.3056867$ ... For $\alpha > \alpha_0$ the series converges, for $\alpha < \alpha_0$ it diverges.

Before the inexpensive computer, it was difficult to tell whether the critical value is ${}> 1$ or not.

Boyd, David W. The sequence of radii of the Apollonian packing. Math. Comp. 39 (1982), no. 159, 249–254.

http://www.ams.org/mathscinet-getitem?mr=658230

added

mentioned in the comments... arbitrarily packed disks, not necessarily touching as in Apollonian packings. The critical value (= dimension of the residual set) is shown to be ${}> 1.02$.

Larman, D. G. On the Besicovitch dimension of the residual set of arbitrarily packed disks in the plane. J. London Math. Soc. 42 1967 292–302.

http://www.ams.org/mathscinet-getitem?mr=209982

This critical value is $\alpha_0 \approx 1.3056867$ ... For $\alpha > \alpha_0$ the series converges, for $\alpha < \alpha_0$ it diverges.

Before the inexpensive computer, it was difficult to tell whether the critical value is ${}> 1$ or not.

This critical value is $\alpha_0 \approx 1.3056867$ ... For $\alpha > \alpha_0$ the series converges, for $\alpha < \alpha_0$ it diverges.

Before the inexpensive computer, it was difficult to tell whether the critical value is ${}> 1$ or not.

Boyd, David W. The sequence of radii of the Apollonian packing. Math. Comp. 39 (1982), no. 159, 249–254.

http://www.ams.org/mathscinet-getitem?mr=658230