Hi. I have a feeling this is kinda a known thing, but I don't know it, so hopefully you can help me out. I'm a lowly software developer.


For a given n and k, is there a short-cut way of calculating (or approximating) the value of this expression? 


$\sum\_{n}^{k}\frac{1}{n}$


It's a matter of computational efficiency: I have a lot of these expressions i want to calculate.



sigma 1/n up to k
If you mean that the sum starts at 1 these are exactly the Harmonic numbers, see <http://en.wikipedia.org/wiki/Harmonic_number>


There are well-known asymptotic expansions (see the above mentioned page for a start, and below).


For actual computation, there is a nice blogpost on computing them by Fredrik Johansson: <http://fredrik-j.blogspot.com/2009/02/how-not-to-compute-harmonic-numbers.html>


