Hey 99.95.

You could show that the series converges by say a ratio test. If you show convergence for all z then you have shown its entire.

As for showing that it satisfies the DE, do a term by term differentiation twice and then show that all terms of the infinite series cancel out when you substitute the derivatives in with the DE equation. (Remember d/dz z^n = nz^(n-1) even when z is complex).