Can I simplify:
\begin{equation} \sum_{x=x_0}^{x_1} \frac{1}{ax+b} \end{equation}
Can I simplify: \begin{equation} \sum_{x=x_0}^{x_1} \frac{1}{ax+b} \end{equation} 


Mathematica (or rather Wolfram Alpha) gives an answer in terms of the digamma function: http://www.wolframalpha.com/input/?i=Sum[1%2F%28a+x%2Bb%29%2C{x%2Cx0%2Cx1}] 


Using some Taylor approximations it turns out there's a fast approximation here. Code below.
I found 


Sometimes I estimate the sum of 1/p, where p ranges over a finite set of prime numbers. I often use 2/a as a lower bound for 1/(a+d) + 1/(ad) when d is small compared to a. I often end up with a handy rational approximation with a small error I can calculate exactly when I need to do so. Perhaps a modification of this idea can help you in making elementary estimates of your sum. Gerhard "Not Quite Splitting The Difference" Paseman, 2012.02.16 

