Why does $\lim\limits_{N \rightarrow \infty}{\sum_{i=1}^{N}\frac{1}{\frac{N}{1-\epsilon}-i}}$ converge to $\log\left[\frac{1}{\epsilon}\right]$?

Question

while playing around with my equations, i found that the following has to hold for my universe to be consistent:

$$\lim_{N \rightarrow \infty}{\sum_{i=1}^{N}\frac{1}{\frac{N}{1-\epsilon}-i}}\rightarrow \log\left[\frac{1}{\epsilon}\right]\text{ for }0<\epsilon<1$$

playing with numerical implementations in mathematica seem to support this by "experiment", but i just don't see why.

Anybody any ideas? Thanks, Martin

Answer 1

Your limit can be seen as a Riemann sum. $$ \lim\limits_{N \rightarrow \infty}{\sum_{i=1}^{N}\frac{1}{\frac{N}{1-\epsilon}-i}} =\lim\limits_{N \rightarrow \infty}{\sum_{i=1}^{N}\frac1N\,\frac{1}{\frac{1}{1-\epsilon}-\frac iN}} =\int_0^1\frac1{\frac1{1-\epsilon}-t}\,dt =-\log\left.\left(\frac1{1-\epsilon}-t\right)\right|_0^1\\ =\log\frac1{1-\epsilon}-\log\left(\frac1{1-\epsilon}-1\right) =-\log(1-\epsilon)-\log\left(\frac\epsilon{1-\epsilon}\right)\\ =-\log\epsilon=\log\frac1\epsilon. $$