Zero entropy change

Question

If you put a object in contact with a heat reservoir that is infinitesimally higher in temperature than the object and allow equilibrium to be reached the entropy change is zero right?

Answer 1

When talking about infinitesimals, you need to specify how close to zero it is. The entropy loss is $\delta Q\over T$ and the gain is $\delta Q \over T-\delta T$, so the net gain is $\delta Q \delta T \over T^2$ to leading infinitesimal order, and it vanishes linearly in $\delta T$ and $\delta Q$ both.

For iterated infinitesimal processes that approximate a path in state space, the entropy gain will zero when you compose it from many nearly-reversible paths, so in this sense, the entropy gain is zero.