# Evaluating $\sum_{k=0}^{{n-1}}\frac1{1-\frac kn}$

I'm working on a problem in probability and got to the sum $\sum_{k=0}^{{n-1}}\frac1{1-\frac kn}$, where $n$ is constant.

I tried changing its form but didn't get anywhere.

Any hint?

• I think you made a mistake in the index of summation. – hamid kamali Dec 12 '15 at 23:33
• Multiply the fraction by $n/n$ and look up "harmonic number". – user147263 Dec 12 '15 at 23:34
• @hamidkamali what do you mean? – Whyka Dec 12 '15 at 23:36
• @NormalHuman oh, that seems interesting. Thank you! – Whyka Dec 12 '15 at 23:36
• consider $\frac{1}{1-\frac{k}{n}}$ when $k=n$. – hamid kamali Dec 12 '15 at 23:38

$$\sum_{k=0}^{n-1}\frac1{1-\frac k n}=\sum_{k=0}^{n-1}\frac n{n-k}=\sum_{m=1}^{n}\frac n{m}=n\sum_{m=1}^{n}\frac 1{m}=n\cdot H_n$$
• I just did this, but wondering how to calculate this $H_n$. Using wikipedia for the meanwhile... Thanks anyway – Whyka Dec 12 '15 at 23:45
• There is no closed form. But it's "roughly" $\ln n$, when $n\to\infty$. – Clement C. Dec 12 '15 at 23:46