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