A polynomial homotopy is a family of polynomial systems, typically in one parameter t. Our problem is to compute power series expansions of the coordinates of the solutions in the parameter t, accurately, using multiple double arithmetic. One application of this problem is the location of the nearest singular solution in a polynomial homotopy, via the theorem of Fabry. Power series serve as input to construct Pade approximations. Exploiting the massive parallelism of Graphics Processing Units capable of performing several trillions floating-point operations per second, the objective is to compensate for the cost overhead caused by arithmetic with power series in multiple double precision. The application of Newton's method for this problem requires the evaluation and differentiation of polynomials, followed by solving a blocked lower triangular linear system. Experimental results are obtained on NVIDIA GPUs, in particular the RTX 2080, P100 and V100. Code generated by the CAMPARY software is used to obtain results in double double, quad double, and octo double precision. The programs in this study are self contained, available in a public github repository under the GPL-v3.0 License.