differential equations - Differentiation of infinite series does not seem to be useful

Trying series solution of differential equations, the routine is to define a function as a series, and differentiate it.

Block[{a, h},
 h[x_] := Sum[a[j] x^j, {j, 0, Infinity}];
 D[h[x], x] // InputForm
]

and I obtained

Sum[j x^(-1 + j) a[j], {j, 0, Infinity}]

I think this is not acceptable in any practical applications.

Actually, when there is an infinite, I can hardly do any further calculations. For example, if I try

Block[{a, h, c},
 h[x_] := Sum[a[j] x^j, {j, 0, Infinity}];
 Series[D[h[x], x] - c h[x], {x, 0, 4}]
] // InputForm

I got

Sum[j x^(-1 + j) a[j], {j, 0, Infinity}] - c Sum[x^j a[j], {j, 0, Infinity}]

Which is not a series expansion. Any idea of how to go further?