# Zero entropy change

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?

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.