Given this sum:
$ \frac{1}{n} + \frac{1}{n+1} + \frac{1}{n+2} + \cdots + \frac{1}{2n-1}$
I am trying to convert (approximate) it to an integral. This is what I have so far:
$ \frac{1}{n} + \frac{1}{n+1} + \frac{1}{n+2} + \cdots + \frac{1}{2n-1} = \frac{1}{n} \left( 1 + \frac{n}{n+1} + \cdots + \frac{n}{2n-1} \right) = \sum_{i=1}^{n-1}{\frac{1}{n}\frac{n}{n+i}}$
How do I continue from here? Also, how do I set the limits of the integral once I find it?
What do I want my sum to look like before I can integrate over it? Are there any conditions?
Thanks