1. Let $\left\{a_{n}\right\}$ be an arithmetic sequence with the sum of the first $n$ terms denoted as $S_{n}$. If $S_{2010}-S_{1}=1$, then $S_{2011}=$
$$
S_{2010}-S_{1}=a_{2}+a_{3}+\cdots+a_{2010}=\frac{2009\left(a_{2}+a_{2010}\right)}{2}=1 \Rightarrow a_{2}+a_{2010}=\frac{2}{2009} \text {, }
$$

Therefore, $S_{2011}=\frac{2011\left(a_{1}+a_{2011}\right)}{2}=\frac{2011\left(a_{2}+a_{2010}\right)}{2}=\frac{2011}{2009}$.