How to calculate this sum of inverses?

Question

I want to calculate this sum:

$$-m u \sum_{i=1}^N \frac{1}{M+im}$$

I tried a substitution like this:

$$-m u \sum_{k=M+m}^{M+Nm} \frac1k$$

But it seems to have become complicated!

Does anyone have an idea how to evaluate the sum?

Answer 1

As Zack Ni commented, this corresponds to generalized harmonic numbers

$$\sum_{i=1}^N \frac{1}{M+im}=\frac 1m\sum_{i=1}^N \frac{1}{\frac Mm+i} =\frac 1m\left({H_{\frac{M}{m}+N}-H_{\frac{M}{m}}}\right)$$