Find change in entropy for a system with a series of reservoirs

[mcas]

Homework Statement

A material is brought from temperature $T_i$ to temperature$T_f$ by placing
it in contact with a series of $N$ reservoirs at temperatures $T_i + \Delta T$, $T_i + 2\Delta T$, ..., $T_i + N \Delta T = T_f$. Assuming that the heat capacity of the material,
C, is temperature independent, calculate the entropy change of the total
system, material plus reservoirs. What is the entropy change in the limit
$N \rightarrow \infty$ for fixed $T_f - T_i$?

Relevant Equations

$dS = \frac{1}{T} Q_{reversibile}$

I've calculated the change in the entropy of material after it comes in contact with the reservoir:

$$\Delta S_1 = C \int_{T_i+t\Delta T}^{T_i+(t+1)\Delta T} \frac{dT}{T} = C \ln{\frac{T_i+(t+1)\Delta T}{T_i+t\Delta T}}$$

Now I would like to calculate the change in the entropy of the reservoir. The answer in the book is:

$$\Delta S_2 = -\frac{C\Delta T}{T_i + (t+1)\Delta T}$$
And I don't know where this answer comes from. How am I supposed to find the change in entropy if I don't know what is the heat capacity of the reservoir?

 

[Chestermiller]

For an ideal constant-temperature reservoir, the change in entropy is always $$\Delta S=\frac{Q}{T_R}$$ where $T_R$ is the reservoir temperature.

 

[mcas]

For an ideal constant-temperature reservoir, the change in entropy is always $$\Delta S=\frac{Q}{T_R}$$ where $T_R$ is the reservoir temperature.

Thank you! I didn't know that.