Is it possible to express $\frac{1}{n}+\frac{1}{1+n}+\dots+\frac{1}{2n-1}$ as a sum
Just to be clear, $\frac{1}{n}+\frac{1}{1+n}+...+\frac{1}{2n-1}$ is already expressed as a sum. What you want is express it with the $\Sigma$ notation. We have $$\frac{1}{n}+\frac{1}{1+n}+\cdots +\frac{1}{n-1+n} = \sum_{k=0}^{n-1} \frac{1}{k+n}$$
I guess you mean to express it in a compact way, since it is clearly a sum. The compact notation is$$\sum \limits _{k=0}^{n-1}\frac{1}{n+k}.$$