# Derivative problem

Hello,
I was trying to take the derivative of the following summation function:
$$f(\varepsilon)=\sum_{n=0}^\infty e^ \frac {-n\varepsilon} {kT}$$

so since the derivative of an exponential function that is raised to a power and that power is function will just be the function itself times the derivative of the power, I figured that the answer would be"

$$\frac {df}{d\varepsilon} = \sum_{n=0}^\infty (\frac{-n}{kT}) . e^ \frac {-n\varepsilon}{kT}$$

and then of course we can take the constants outside the summation

but to my surprise, the book did it this way:

$$\frac {df}{d\varepsilon} = \sum_{n=1}^\infty (\frac{n}{kT}) . e^ \frac {-n\varepsilon}{kT}$$

so my question is how did the lower limit change from n=0 to n=1 and where did the minus sign (-n/kT) go? do these two have something to do with each other?

thanks a lot

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