I am not 100% confident about this so would someone mind checking my work?

The nth partial sum of a Series is given by:

$\displaystyle Sn=\frac{1}{n+1}+\frac{1}{n+2}+\frac{1}{n+3}+...+\ frac{1}{2n}=\sum_{k=1}^{n}\frac{1}{n+k}$

1. What is the associated Series:

$\displaystyle \sum_{n=1}^{\infty}\frac{1}{2n}$

2. Convergence?

$\displaystyle \lim_{n\to\infty}\frac{1}{2n}=0$

General term goes to 0

Test for convergence by direct comparison.

$\displaystyle \frac{1}{2n}=\frac{1}{n-\frac{1}{2}}>\frac{1}{n}$

$\displaystyle \frac{1}{n}$ divergent harmonic series

$\displaystyle \sum_{n=1}^{\infty}\frac{1}{2n}$ diverges by direct comparison with $\displaystyle \frac{1}{n}$

3. Determine the Limit if possible

$\displaystyle \lim_{n\to\infty}\;Sn=\lim_{n\to\infty}\frac{1}{2n }=0$