And I can't figure out how to get this. I got T(0,t) = 100, therefore A1 e^(-c^2 t) = 100, but that doesn't tell me much. I know I need to sum over c or n at some point, but am I right in thinking you can't sum over c yet because it's a real arbitrary constant, rather than an integer n? That would probably give you the 1 though from n=0. But the problem is with the insulated end x=L, surely if it's insulated it won't lose heat, so the temperature would just go up?

The maximum temperature at any point is 100 degrees. The temperature at the insulated end will only reach this limit at t = infinity. So you have T(L,inf) = 100.