Sorry for the vague question name, since I am looking for the name of the series. Also this might not be a "series" by the strict definition of a series.. anyways here it is:
Choose some $m$ and $n$ as some positive integer, and $m> n$. $$\sum_{i=0}^n \frac{1}{m-i}$$
Like Kundor stated above, this is a harmonic sum with strange bounds. (like you stated)