a service economy is unsustainable
Why?
It's simply the LR trend to become a service-oriented economy (at least theoretically).
As productivity in agriculture increases it frees up labor for manufacturing industries and once productivity increases in manufacturing industries, labor is again freed up to work in service industries. This can be seen also to correlate to the rural => urban => suburban transformation throughout our economic history (and other countries as well).
In terms of rate of productivity growth as defined by a production fn in the form of f(K,L) that would imply Q/L=A, we see that in terms of the rate of growth (conceptually think like a second derivative, x''). Agriculture > Manufacturing > Service.
This is pretty much confirmed by the graph that Beet posted, showing the overall value of manufacturing output to increase (which is due to two things 1) productivity increases and 2)"moving-up-the-value-added-ladder").